# User:Garrett E. McMath/Notebook/Junior Lab/2008/11/17

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# Estimating Planck's Constant via the Photoelectric Effect

• Author: Garrett McMath
• qbgem@unm.edu
• Experimentalists: Garrett McMath and Paul Klimov

## Abstract

Planck's Constant is one of the most fundamental quantities in all of physics, and essential for quantum mechanics. In this experiment we set up a way to measure this fundamental value with reasonable accuracy. We used a commercial apparatus to estimate Planck's constant via the photoelectric effect from varying frequencies of light. In so doing we were able to achieve results that correlated with the accepted value with only 9% error. Given the rudimentary nature of the lab setup we felt this was a very good estimate of Planck's constant given the equipment we had to work with.

## Equations

$Vo=\frac{h}{e}f - \frac{W_{o}}{e}$

• (where Vo is the stopping potential, e is the fundamental electron charge, f is the frequency of the photon, h is Planck's constant, and Wo is an intrinsic property of the material called the work function)

to determine Planck's constant h. This is done by the fact that when Vo and f are graphed in Cartesian coordinates they will produce a straight line with a slope equal to Planck's constant and an intercept equal to the work function of the material.

## History

Max Planck, the originator of Planck's Constant, was a German physicist who lived from 1858 to 1947. In the year 1900 he announced that by making some then strange assumptions he had found a solution to the problem of the Rayleigh-Jeans equation at short wavelengths. The Rayleigh-Jeans equation was used at the time to describe the energy density distribution function of light. This equation was very accurate at long wavelengths but experimentally fell apart at shorter wavelengths. This was due to the fact that the equation had been derived using classical mechanics, that is that the average energy per mode of oscillation is same as a harmonic oscillator in one dimension. The problem is that this predicts that as the wavelength approaches zero the energy density distribution goes to infinity while experiments had shown it actually approaches zero. Planck remedied this problem by assuming that the average energy of the oscillating charges and conversely the radiation they produce was a discrete variable meaning they had to take values of 0,ε,2ε,3ε,...nε where n=0,1,2,3... This meant that Planck could now write the energy as En = n = nhf. This idea that energy was quantized spurred quantum mechanics/modern physics as we know it. The actual idea was actually brought to full fruition by Einstein in 1905 when he used Planck's ideas to both explain the photoelectric effect as well as postulate that this quantization was a fundamental characteristic of light and not some mysterious property of oscillators in cavity walls.[3]

## Introduction

This experiment relies heavily on the physics of the photoelectric effect. The photoelectric effect, discovered by accident by Heinrich Hertz(1857-1894) in 1887, was explained by Einstein in his Annalen der Physik[2]. Einstein stated that the energy quantization used by Planck in solving the Rayleigh-Jeans problem (known as the ultraviolet catastrophe) defined light energy as discrete quanta each with energy hf.[3] These quanta, known as photons, can be completely absorbed by electrons. With enough energy, a photon can raise the energy of the electron that absorbs it so much that it is ejected from the metal with some kinetic energy. The minimum amount of energy needed to eject an electron is known as the work function and is specific to the metal the electron is in. Therefore the maximum kinetic energy of an ejected electron is the energy of the photon it absorbed(hf) minus the work function of the metal. The main equation that allows us to measure Planck's constant is obtained by setting this maximum kinetic energy equal to a potential multiplied with the fundamental charge of an electron. When the two sides of the equation are equal that potential is what is called the stopping potential because in the experiment what is measured is the voltage difference caused by the ejected electrons. The electrons are ejected into a negative potential difference which doesn't allow any electrons to reach the anode unless they have a kinetic energy greater than the potential. Thus if the energy difference is less than the stopping potential no electrons will reach the anode. Modern determinations of Planck's constant are extremely accurate and are far more advanced than the scope of this lab. The modern accepted value as recorded by the National Institute of Standards and Technology (NIST) is obtained primarily from a method known as the Watt Balance. A Watt Balance is an apparatus that measures two powers one in watts and the other in standard electrical units which produces the measure of their product. Using a value known as the von Klitzing constant (used in quantum Hall effects) this produces a direct measure of Planck's Constant from the equation[4], where RK = h/e2,

$h = \frac{4}{K_{\rm J}^2 R_{\rm K}}$

It is worth noting however that before all the new age physics, Planck managed to calculate the value to within 1.2% of today's accepted value using nothing more than blackbody radiation data and some statistical mechanics[7]. While our experiment had no chance of duplicating the accuracy of a the Watt Balance, We measured Planck's constant using the much of the same physics as were used in its original determination.

## Procedure/Setup

Figure 1: PASCO h/e Apparatus Setup[1]
Figure 1: Schematic diagram of PASCO h/e Apparatus[1]

The overall goal of this lab was to establish the quantum nature of light and to accurately measure Planck's Constant. This was accomplished through the following experiments.

1. Measurement of stopping potential and charge time with variable light intensity
2. Measurement of stopping potential with variable light frequency
3. Measurement of stopping potential with variable order and frequency

All the equipment used for the experiments were PASCO apparatuses, except for a Wavetek 85XT RMS DVM multimeter, and can be seen in the PASCO Manual for Apparatus[1] along with the procedure followed for each experiment. Each experiment proved different aspects of the quantum nature of light while the measurement of Planck's Constant was found through data analysis of the data from the 2nd and 3rd experiments.

## Experiments

Figure 1a: Intensity of yellow light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the yellow spectrum of light through. (Excel is a Microsoft® based application)
Figure 1b: Intensity of green light vs. mean charge time. Three different trend lines and their R-squared values have been added: linear, exponential, and power. Intensity was was controlled by a filter with five sections corresponding to 20%, 40%, 60%, 80%, and 100%. The filter used computer generated dots to block out a certain amount of light. A filter was also used to allow only the green spectrum of light through. (Excel is a Microsoft® based application)

#### Experiment 1-Qualitative

Experiment 1 proved the non classical result that the stopping potential i.e. the maximum kinetic energy of the electrons is not related to intensity. At each of the five intensities the stopping potential was unaffected other than the small charge leak due to the non infinite impedance in the apparatus. The experiment also proved that the charge time is affected by intensity. We know from physics that the affect on charge time should be linear however as seen in the figures 1a and 1b ours was more exponential we believe this is due to the charge leak, and we could not find any reasonable information on the apparatus to allow us to calculate this loss and therefore was not adjusted. The overall conclusion was the intensity of light affects how many electrons are ejected from metal but not the speed they are ejected at.

#### Experiment 2 and 3-Quantitative

Experiments 2 and 3 involved the measuring of the stopping potential of the different spectra of mercury in both the first and second order spectra. Data analysis provided the Planck's constant and work function. Comparing the first and second order provided evidence that the green spectra in the second order was being overlapped/corrupted with another band of light. The PASCO manual confirmed this suspicion stating that the frequency of the second order green band is interfered by an ultraviolet band of the third order.

Figure 2a: Mean stopping potentials for each interference fringe are shown, with the SEM of each point represented by error bars. A least squares line is fit to the data without constraints. In addition, a slightly modified line is also shown, which was generated by moving the least squares line within, or in close proximity, of the error bars. (Excel is a Microsoft® based application)

## Data Analysis

• Work function: 1.36(08)eV[6]
• Plank's Constant: 4.13566733(10)E^-15eV*s[5]

Accepted values of Mercury spectra

Yellow Green Blue Violet Ultra-Violet

Frequency(Hz)

5.18672E14 5.48996E14 6.87858E14 7.40858E14 8.20264E14
Wavelength(nm) 578 546.074 435.835 404.656 365.483
• (Wavelengths[1], frequencies calculated using c/λ (c=speed of light, λ=wavelength))
Figure 3: Standard Deviation vs Frequency, A plot of the standard deviation of each point plotted against the frequency corresponding to that point. A very rough trend showing that the data got less precise as the frequency got higher. (Excel is a Microsoft® based application)

The analysis of the data was done in Excel (Excel is a Microsoft® based application). The analysis involved taking averages of our stopping potential data, since we performed the experiment several times for each frequency, and making a scatter plot of the average stopping potential vs the corresponding frequency of light. A best fit linear line was was put on the graph and its R-squared value to show the linearity of this relationship. The actual Planck's constant and uncertainty values were calculated using Excel's LINEST function on the average stopping potential and frequencies. The results are as follows:

Accepted Uncertainty Experimental Uncertainty Percent Error

Planck's Constant

4.13566733E-15 eVs 2.5E-8 eVs 4.54234E-15 eVs 5.13689E-17 eVs 9.83%
Work Function (-)1.36 eV .o8 eV (-)1.63573557 eV .034576054 eV 20.27%
• (Accepted value for Planck's Constant[5], Accepted value for work function[6])

Clearly we did not achieve the optimal results. Our data did not encompass the accepted values even at the most extreme ranges of both uncertainties. However, given the systematic error we could not account for (i.e. the charge leak and multimeter errors) the data was within our expectations. Figure 3 show the standard error of the data sets vs the frequency they were taken at. It shows a very rough trend that at higher frequency we got larger standard deviations meaning our data was getting less precise. Obviously the experiment must be done differently or with a correction for the charge loss in order for the data to within any reasonable confidence interval.

## Conclusions

This lab was simple in nature and because of that it was very possible to eliminate human sources of error. However the price you pay is its nearly impossible to correct for the systematic error inherent in the apparatus. The only errors that make a significant difference in this lab are the charge leak from the op-amp in the apparatus and the always present multimeter correction for non infinite impedance. I attempted a side experiment in an attempt to find a way to correct for the charge leak. The idea behind the experiment was that given longer times of charging, the leak would become more and more a factor. Therefore it seemed that by doing the stopping potential experiment at different intensities and graphing the results would produce a line whose slope would be used as a correction factor for the leak. The experiment failed to give such a result. I did not have sufficient enough time to explain the results, but they were basically the exact opposite of what we had hoped would happen. The percent error in the calculation of Planck's constant actually improved when less intensity was used in the data acquisition, instead of worsening as hoped. I could find no plausible physics reason for this, it seems to be part of the systematic error of the lab. The conclusion reached is that there is unequivocal evidence for the quantum nature of light and that given an accurate measure of Planck's Constant, we can describe the quantization of the energy in that light. These conclusions though somewhat elementary in this day and age are still some of the most profound and influential to modern physics as we know it.

## Acknowledgments

My thanks to Paul Klimov for his help in performing these experiments and in the valuable information gained from our discussions of the results. Also much appreciation must be given to Dr. Steven Koch and Aram Gragossian for their help in understanding the safety aspects of the lab as well as invaluable information gained from talking about the inner workings of the PASCO apparatus.

## References

1. Ayar, Eric. Griffith, Dave (editor). h/e apparatus and h/e apparatus accessory kit. Instruction Manual and Experiment guide for the PASCO scientific Mode AP9368 and AP 9369. Roseville, CA: PASCO Scientific. 1989.
2. Einstein, Albert. On a Heuristic Viewpoint Concerning the Production and Transformation of light. Annalen Der Physik 17: 1905.
3. Llewellyn, Ralph A;Tipler, Paul A. Modern Physics. 5th ed New York, NY: W.H. Freeman and Company. pp119-132. 2008.
4. Mohr, Peter J; Taylor, Barry N; Newell, David B. CODATA Recommended Values of the Fundamental Physical Constants. Reviews of Modern Physics 8:633-730. 2006
5. Physics Laboratory. Jun.1994/Apr. 2008 National Institute of Standards and Technology. 14 Dec. 2008 <http://physics.nist.gov>
6. Tech Note 303 Detail. 5 Dec. 2001/10 Oct. 2003. PASCO 10 Dec. 2008 <http://www.pasco.com/support/technical-support/technote/techidlookup.cfm?technoteid=303>
7. Planck, Max. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik 4: 553. 1901