Planck's Constant and the Photocell

 

An apparatus measures Planck's Constant using a photo-electric cell. The complications that limit its accuracy are discussed in detail.

The Photoelectric Effect and Planck’s Constant

When light shines on the surface of certain metals, electrons can be ejected in a process known as the Photo-Electric (PE) effect. This can be used to create light detectors and offers insight into the fundamental nature of light. 

Light is often described as bundles of energy called photons. Each photon has an energy value $E$ that depends only on its frequency $f$ as $E = hf$, where $h$ is Planck’s constant. Light frequency can also be written as $f = \frac{c}{\lambda}$, where c is the speed of light and $\lambda$ is the wavelength. The photon energy is then $E = \frac{hc}{\lambda}$. When a photon strikes a metal surface, an electron may absorb its energy. If this newly acquired energy is greater than the binding energy of the surface $W$ (workforce), then the electron will be ejected and travel with the remaining kinetic energy $E’ = \frac{hc}{\lambda} – W$. 

The Photoelectric Tube (Vacuum Photocell)

In a vacuum photo-cell, a curved sheet of metal (photo-cathode) is placed near a wire electrode (anode) in a vacuum. When the cathode is attached to the negative side of a voltage source and the anode to the positive side, electrons ejected from the cathode by light accelerate toward the anode, establishing an electrical current. The applied voltage V imparts energy eV to the photo-electrons, where e is the electron charge. This gives the photo-electron a total energy of $E’’ = \frac{hc}{\lambda} – W + eV$. As the electrons flow from the cathode to the anode, a positive current is said to flow (forward bias case shown below).

If a positive voltage is applied to the cathode, the electron energy E’’ will be reduced and the photo-cell is said to be reverse biased (middle figure above). In principle E” = 0 at some value of $V = V_s$, known as the stopping voltage, where $eV_s = \frac{hc}{\lambda} – W$. This condition should be detectable as zero current flow if all the electrons are stopped by the reverse voltage. If we could measure $V_s$ for various incident wavelengths, then a plot of $V_s$ vs. $\frac{1}{\lambda}$ should have a linear slope of $\frac{hc}{e}$ and an intercept of $–\frac{W}{e}$. Other experiments tell us that $c = 3 \times 10^8 m/s$, and $e = 1.6 \times 10^{-19}$ coulombs, allowing us to calculate Planck’s constant $h$ from the slope and work function $W$ from the intercept.

Circuit Design

Goals:

• Construct a circuit to apply a bipolar bias voltage to a photo-cell.
• Provide light sources of known wavelengths.
• Provide a means to monitor the voltage and current flow.

The Circuit, consisting of a photo-tube, battery, potentiometer and polarity switch:

Notes:

• A 10 turn 50K $\Omega$ potentiometer is used to vary the voltage, and the polarity is controlled by a double-pole double throw (DPDT) switch. These components are placed into a black plastic enclosure.
• $V_{bias}$ and current are measured by an external voltmeter and ammeter.
• The light sources were either laser diodes or LEDs.

Experiment Configuration

Main Circuit Box

Pico-Ammeter 

The goal is to apply a voltage $V_{bias} = V_s$ that results in zero current for each illumination wavelength. Detecting low current levels requires a current meter with sensitivity in the pico-amps ($10^{-12}$ amps) near $V_s$. Most inexpensive multi-meters read only as low as milli-amps. Fortunately, there are designs for pico-ammeters that can be constructed for low cost. 

The basic circuit for an ammeter (see the above diagram) is just a resistor R with a voltage sensor connected across it. Ohm’s law tells us that for a voltage V across the resistor, the current I passing through the resistor is $I = \frac{V}{R}$. When $I = 10^{-12}$ amps, a voltage reading of 1 milli-volt requires $R = 10^9 \Omega = 1 G\Omega$.

Low current circuits use very high amplification and require short leads with low inductance to avoid spurious oscillation. Direct solder between components and wide conductive planes are usually recommended. Best practices are discussed in detail in this article:   https://www.nutsvolts.com/magazine/article/may2011_fattaruso 

The recommendations include:

• Copper on PCB cut using a Dremel tool to define the ground, V+ and V- regions. 
• Jewelry crimps for soldering to IC pins pointing out from the board. Other IC leads were soldered directly to the pads. 
• The PCB bolted to an Al box for grounding.
• Capacitors at 0.1uF soldered across the pads to dampen stray oscillation. 
• Desiccant placed in the box to reduce moisture that could otherwise cause spurious paths for current.

Ammeter Schematic and Construction:

Reference circuit design is from stoppi, an excellent source for many hobbyist high-end physics experiments. https://stoppi-homemade-physics.de

Testing the ammeter

• Test of the pico-ammeter: A 1.5V battery and $10K\Omega$ potentiometer were used to make a variable voltage $V_{src}$, fed through a $10M\Omega$ resistor ($R_s$) to provide a variable current $I_{src} = \frac{V_{src}}{R_s}$ into the current amplifier. The output voltage $V_{src}$ was read with a multi-meter, while $V_{amm} = I_{amm}R_{gain}$ from the amplifier was read by another multi-meter. The plot below shows that $I_{amp} =  I_{src}$ for $R_{gain} = 1 G\Omega = 1000M \Omega$, as was also shown for each $R_{gain}$ value.
• Note that with a 12V battery for a split power supply, the maximum current that can be detected is $6V/ R_{gain}$. For example, at the 1 pA setting, $R_{gain} = 1G\Omega$ and the maximum measurable current is 6000 pA.
• At the 1 pA setting, it is crucial to minimize user motion near the amplifier input connections as this can induce stray currents. A good procedure for this setting is to wait about 30sec for each reading to stabilize.

Light Sources

• Each stopping voltage $V_s$ is associated with a specific wavelength for the measurement of $h$. 
• Most demonstrations of this experiment use an expensive bright white light source with color bandpass filters that produce a somewhat wide bandwidth of wavelengths (narrow band is preferred). 
• In this project, four laser diode sources and six LEDs were used with peak wavelengths ranging from ultra-violet to red.
• For convenience, the lasers and LEDs were mounted on a turret whose axis was bolted to the main circuit box, described below.

Laser light sources

• Turret design using a selector switch as the axel.
• Rotating the turret activates the laser when rotated to face the light entrance.
• All laser modules were obtained for < $20 ea. Note that some module housings are not grounded and must be electrically insulated.

Laser Wavelength Measurements

(Using Spectrometer Project with motorized scan)

LED light sources

• Turret design using a selector switch as the axel.
• Rotating the turret activates the LED when rotated to face the light entrance.
• All LEDs were clear 5mm diameter.

LED Spectra (Normalized to Peak)

Note: These LED spectra are far broader in wavelength than the laser diodes. This can broaden the range of stopping voltages needed to achieve zero photo-current.

Neutral Density Filters

• To reduce the intensity of the light sources as needed, optical Neutral Density Filters (NDF) were purchased from Thor Labs https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=5011 , and were placed between the sources and the photo-cell. 
• The transmission T of an NDF is measured by its Optical Density (OD), where $T = 10^{-OD}$. OD = 0.3, 3, 4 and 5 were used. 
• Each filter has a specified transmission spectrum, as shown below for OD3 and OD4*. The laser wavelengths are indicated by the red lines, and the LED peak wavelengths by the orange arrows.

*Note: the Transmission of the OD3 NDF was confirmed to be 10 times that of OD4 at all wavelengths from 400nm to 670nm

Vacuum Photo-Cells

• The ideal photo-cell would be constructed from a single metal cathode with a specific work function W and no conductivity between the cathode and anode. One example would be a Leybold cell made with metal coated by pure potassium, protected in a vacuum tube, where the anode is a thin platinum wire. These are typically very expensive.
• A much lower cost option is to use a vacuum photo-cell made in the 1940s by companies like RCA, available from antique electronics supply houses. This was the choice used in this project for cost reasons, but it did lead to non-ideal performance issues.

Photocells (RCA 923)

• One example phototube is RCA 923 made approximately 1940 - 1950. Tube Source was: https://www.radiomuseum.org/tubes/tube_1p37.html
• The photo shows the backside of the cathode with the vertical anode wire behind it. Light enters from the anode side.
• The tube is specified as “gas filled”, presumedly at a low pressure that allows transport of  photo-electrons.
• This tube has an S-1 spectral response from the manufacturer (shown below) with peaks at 800nm and 360nm wavelengths. This may indicate that the cathode contains multiple metals with multiple values for $W$ that could make it difficult to derive a value for $h$ from the data.

Photocell (RCA 1P37)

• Another example of a phototube is the RCA 1P37 also made approximately 1940 - 1950. Tube Source was: https://www.radiomuseum.org/tubes/tube_1p37.html
• The photo shows the front of the cathode with the anode wire pointed vertically in front of it. Light enters through the glass onto the cathode.
• The tube is also specified as “gas filled”, presumedly at a low pressure that allows transport of photo-electrons.
• The spectral response from the manufacturer is S-4 type, as shown below, measured using equal intensity spectral light input. The response peaks at a wavelength of 400nm. The vertical red lines indicate the laser wavelengths, and the orange arrows show the LED peak wavelengths used in this experiment. The photo-current depends on the light intensity times spectral response, and both are a function of wavelength.
• This tube was chosen for the project due to the single peak response. 

Current vs. Voltage Measurements

Photo-cell Dark Current ($I_d$)

Ideally, a phototube consists of two isolated electrodes that only generates a current when light strikes the photocathode. In practice the RCA 1P37 used in this project produced a measurable current when light is blocked. This is often referred to as Dark Current, $I_d$. Measurements showed $I_d$ varied linearly with the applied voltage (plotted below in blue) that suggests a leakage resistance $R = \frac{V}{I_d}$ that can result from the manufacturing process or aging, which weakly connects the cathode and anode. Any measurement of current from the photocell should subtract out $I_d$ to obtain the true photo-current.

The data indicates that $I_d$ for this tube, measured in picoamps (pA) can be modeled as:

 

$I_d = 22.4 V_{bias}$

This is equivalent to a resistor of $R = \frac{1V}{22.4 pA} = 45 G\Omega$. The dark current was also measured without the tube in place (orange dots) to show that the tube itself is the source of dark current leakage.

To determine the $V_{bias}$ value at which the photo-current is zero (to determine $h$), it is necessary to measure the current vs. $V_{bias}$ and subtract the $I_d$ function.

Sources of Photo-Current

Photoelectrons are generated when light of energy $hf$ greater than a metal’s work function $W$ ejects an electron. The largest metal surface in the phototube is the cathode, whose $W$ value is lower than the energy of visible light or UV. Light can also strike the smaller anode metal (center axial wire), but these are usually made from higher work function metals. If a portion of the metal deposited onto the cathode contaminates the anode in manufacturing, its work function can be reduced, resulting in an anode photo-current that will add to the cathode current when in reverse bias.

A measurement of the phototube current vs. bias voltage is shown below using the blue laser source ($\lambda = 446nm$). The orange curve shows the case for the laser directed only at the cathode. The blue curve shows the case for the laser mostly striking the wire anode, showing a strong negative shift in current. This results in a considerable increase in the voltage required for zero current.

In all cases for the laser sources, care was taken to avoid the beam striking the anode. For the LEDs, the light is diffuse, and some portion always strikes the anode.

Current vs. Voltage for Photocell RCA 1P37 + Laser Sources

Laser sources striking only the cathode were used for the data shown below. The bias voltage was scanned from -3.2 to +3.2 volts and the current (minus $I_d$) was recorded. Each case shows low current at the most negative voltage, a quick rise at a first knee point, a linear portion, and a second knee indicating current saturation. The first knee point occurs at less negative voltages as the wavelength increases. These are typical characteristics in vacuum photocells (see references at the end). 

For three of the lasers, two levels of NDF transmission (OD) were used, differing by an order of magnitude in transmission. The lower transmission OD5 data is multiplied by 10 to compare to the OD4 data, showing general agreement in the curve shapes over a 10x intensity change (the low phototube response to 653nm and the lower brightness of the laser precluded using a 10x drop in intensity). 

Current vs. Voltage for Photocell RCA 1P37 + LED Sources

LED sources diffusely illuminating the anode and cathode were used. The bias voltage was scanned from -3.2 to +3.2 volts and the current (minus $I_d$) was recorded. The curves mimic the shape shown for the laser sources. The first knee point occurs at less negative voltages as the wavelength increases. 

Determining the Stopping Voltage ($V_s$)

From the basic analysis of the photo-electric effect, a photon ejects an electron resulting in an energy $hf – W$ and the photo-current should decrease to 0 when the bias voltage satisfies $eV_s = hf – W$. It should remain 0 as the bias voltage becomes more negative, creating a sharp first knee point, but this is not observed. In reality, there are several other factors at work, which complicate determining $V_s$ from the current-voltage (IV) curves.

• The electron energies in the metal and the work function that binds them is actually a distribution of energies that is temperature dependent.
• The composition of the cathode metal may include multiple metals and work functions.
• Counter current from anode photoelectrons can occur.
• The shape of the first knee near $V_s$, which transitions from a linear slope to near 0 slope, defies using a simple zero-crossing definition for $V_s$, often used for this experiment in student laboratories.

References cited at the end of the post discuss the difficulty in determining $V_s$ due to these complications. Results for $h$ are often underestimates and can vary by a factor of two from the accepted value. 

In these measurements, the least ambiguous method was to use the voltage of the first knee point for $V_s$.

Determining $V_s$ (RCA 1P37 + Laser sources)

Data is plotted below from two runs using an OD4 NDF for the laser sources, near the low current region after the dark current is subtracted. The region where the photo-current is essentially zero is very flat, making a zero-crossing point difficult to accurately determine.

The method used in this study is to find the voltage of the first knee point by fitting it to a circle (indicated in the upper right plot). 

Results

The values of $V_s$ vs. light peak frequency are shown, using the first knee point to determine $V_s$:

For the laser sources the average slope is $\frac{h}{e} = 3.06 \times 10^{-15}$ (kg $m^2$/sec coulomb), and the intercept is $W$ = 1.34 (eV), resulting in $h = 4.9 \times 10^{-34}$ (kg $m^2$/sec). The accepted value for $h = 6.626 \times 10^{-34}$ (kg $m^2$/sec or Joules sec), which is within 26% of this result. 

For the LEDs, the slope is $\frac{h}{e} = 2.09 \times 10^{-15}$ (kg $m^2$/sec coulomb), and intercept is $W$ = 0.67(eV), considerably lower than the laser results. The LEDs differ by their broader bandwidth that could widen the range of $V_s$, and by their diffuse output that contributes a reverse photo-current from the anode, which could invalidate the $V_s$ values. 

References/ Summaries

Student Lab Instructional Sources

Commercially available demonstrations of the photo-electric effect are utilized in educational institutions to introduce the basic effect and to estimate $h$. For the purpose of simplicity, for those that utilize photocells, some of the complexities discussed in this post are often not addressed in their procedures.

Experiment 6 - The Photoelectric Effect | UCLA Physics & Astronomy

Example sites:

https://iopscience.iop.org/article/10.1088/1742-6596/2377/1/012076

http://instructor.physics.lsa.umich.edu/adv-labs/Photoelectric_Effect/photoelectric_effect_w06.pdf

https://demoweb.physics.ucla.edu/content/experiment-6-photoelectric-effect

https://users.rowan.edu/~klassen/dpa/current/ModernLabs/PlancksConstantPhotoeffect.pdf

http://www.physlab.iitkgp.ac.in/Manuals/9_PhotoelectricEffect.pdf

https://www.niser.ac.in/cmrp/storage/labManuals/Planck_s constant.pdf

http://www.cas.miamioh.edu/~marcumsd/p293/lab1/lab1.htm

Stoppi – Homemade Physics

https://stoppi-homemade-physics.de/photoeffekt/

(experiment 2)

• Utilizes photocell tube RCA 1P37 (same as this project)
• LED sources were used
• Stopping voltage ($U_{gegen}$) is measured for zero current (dark current is not explicitly discussed)
• Obtained slope of $V_s$ vs. f to obtain $\frac{h}{e} = 3.1 \times 10^{-15}$ kg $m^2$/sec coulomb and $h = 5 \times 10^{-34}$ kg $m^2$/sec

A. Melissinos, Experiments in Modern Physics - 1st edition 

(Academic Press, 1966), chapter 1 section 4

• Used Leybold tube with a potassium coated cathode, a mercury lamp with color filters, and a sensitive current meter (electrometer or pico-ammeter).
• “Photo emission from the anode introduces difficulties in the exact determination of the stopping potential [$V_s$].”
• “Dark current must be monitored and appropriate corrections applied.”
• Values for Vs were extracted from the current voltage curves using a) the intersection of the tangents of the knee portion, and b) the onset voltage of the sudden rise in current. Both methods gave the result in the slope of $V_s$ vs. f, giving $h = 6.14 \times 10^{-34} Js$. The value for $W$ was less than the accepted value for potassium.