Photoelectric effect

Theory

In 1902, Philipp Lenard, an assistant to Heinrich Hertz, used a high-intensity carbon arc light to illuminate an emitter plate. Using a collector plate and a sensitive ammeter, he was able to measure the small current produced when the emitter plate was exposed to light. In order to measure the energy of the emitted electrons, Lenard charged the collector plate negatively so that the electrons from the emitter plate would be repelled. He found that there was a minimum “stopping” potential that kept all electrons from reaching the collector. He was surprised to discover that the “stopping” potential, $V$ (and therefore the energy of the emitted electrons) did not depend on the intensity of the light. He found that the maximum energy of the emitted electrons did depend on the color, or frequency, of the light.

Three years later, Einstein came up with an explanation for Lenard's observation (in fact, it was for this theory in particular that Einstein received the Nobel prize in 1921). According to Planck's theory, the energy of particles is quantized. Thus, each of the photons arriving at the plate carries an energy $E=h\nu$, where $h$ is Planck constant and $\nu$ is the frequency of the light. On the other side, for an electron current to be measured we need to provide the electrons with:

enough energy to break their bond to the solid (let's call it $W_0$), and
some kinetic energy $K_E$ that allows them to move from their initial position to the ammeter.

The energy transfer takes place as an elastic collision: the energy of the photon is completely transferred to the electron (we could say that the photon was absorbed by the solid). Therefore we have \begin{equation} \label{eq:equation1} h\nu = W_0 + K_E\,, \quad \mathrm{or\,equivalently} \quad K_E = h\nu -W_0\,. \end{equation} Naturally, the expression on the right of Eq. (\ref{eq:equation1}) is only valid when the energy of the impinging photon is larger than $W_0$. Finally, the connection between the kinetic energy of the electrons and the observed current (or voltage difference) is obtained by noting that the energy of electrons in motion is $K_E = -e V$, with&bnsp;$e$ the charge of the electron and $V$ the voltage difference. Thus, to stop the current (as Lenard did in his experiment), the required stopping potential is then $-V$.

Solving Eq. (\ref{eq:equation1}) for the potential, we obtain \begin{equation} V = \frac{h\nu}{e} - \frac{W_0}{e}\,, \end{equation} which is linear with respect to the frequency and independent of the intensity of the light (in agreement with Lenard's observation). Thus, measuring the stoppig potential as a function of the frequency of imping light allows us to obtain:

the ratio $h/e$ as the slope of the line, and
the ratio $W_0/e$ as the intercept of the line in the limit $\nu \rightarrow 0$.

$W_0$ is referred to as the work function and it dictates the energy with which electrons are bound to the solid. Naturally, the work function varies for different materials.

Procedures

Measuring Planck's constant

The setup is arranged following the instructions provided by Pasco.

Initial configuration

Cover the window of the Mercury Light Source Enclosure with the Mercury Lamp Cap. Cover the window of the Photodiode enclosure with the Photodiode Cap.
Adjust the distance between the Mercury Light Source enclosure and Photodiode enclosure so that the general spacing is between 30.0 cm to 40.0 cm. NOTE: The recommended distance is 35.0 cm.
On the Mercury Lamp Power Supply, press the button to turn on MERCURY LAMP. On the Tunable DC (Constant Voltage) Power Supply and DC Current Amplifier, push in the POWER button to the ON position.
Allow the light source and the apparatus to warm up for 10 minutes.
On the Tunable DC (Constant Voltage) Power Supply, set the Voltage Range switch to $-4.5\,\mathrm{V}$ – $0\,\mathrm{V}$. On the DC Current Amplifier, turn the CURRENT RANGES switch to $10^{-13}\,\mathrm{A}$.
On the DC Current Amplifier, push in the SIGNAL button to the “in” position for CALIBRATION.
Adjust the CURRENT RANGES knob until the ammeter shows that the current is zero.
Press the SIGNAL button so it moves to the “out” position for MEASURE.

Measurements

Gently pull the aperture dial away from the case of the Photodiode Enclosure and rotate the dial so that the 4 mm diameter aperture is aligned with the white line. Then rotate the filter wheel until the 365 nm filter is aligned with the white line. Finally, remove the cover cap.
Uncover the window of the Mercury Light Source. Spectral lines of 365 nm wavelength will shine on the cathode in the phototube.
Start previewing in Capstone and click the first row in the table display
Adjust the VOLTAGE ADJUST knob on the DC Power Supply until the digital meter on the DC Current Amplifier shows that the current is zero.
Press “Keep Sample” on the Sample Control bar to record the magnitude of the stopping potential for the 365 nm wavelength in the table display.
Rotate the filter wheel until the 405 nm filter is aligned with the white line. Spectral lines of 405 nm wavelength will shine on the cathode in the phototube.
Adjust the VOLTAGE ADJUST knob on the DC Power Supply until the digital meter on the DC Current Amplifier shows that the current.
Click the second row of the table display and press “Keep Sample” to record the magnitude of the stopping potential for the 405 nm wavelength in table display.
Repeat the measurement procedure for the other three filters. Record the magnitude of the stopping potential for each wavelength in the table, and then press “Stop” in the software.
Repeat all the measurements for the 2 mm and 8 mm apertures.
Turn off the MERCURY LAMP power switch and the POWER switch on the other pieces of equipment. Rotate the filter wheel until the 0 nm filter is aligned with the white line. Cover the windows of the Mercury Light Source Enclosure and Photodiode Enclosure.

Measuring the dependence on the intensity

Initial configuration

Cover the window of the Mercury Light Source enclosure with the Mercury Lamp Cap. Cover the window of the Photodiode enclosure with the Photodiode Cap.
Adjust the distance between the Mercury Light Source enclosure and Photodiode enclosure so that the general spacing is between 30.0 cm to 40.0 cm. NOTE: The recommended distance is 35.0 cm.
On the Mercury Lamp Power Supply, press the button to turn on MERCURY LAMP. On the Tunable DC (Constant Voltage) Power Supply and DC Current Amplifier, push in the POWER button to the ON position.
Allow the light source and the apparatus to warm up for 10 minutes.
On the DC (Constant Voltage) Power Supply, set the Voltage Range switch to $-4.5\,\mathrm{V}$ – $30\,\mathrm{V}$. On the DC Current Amplifier, turn the CURRENT RANGES switch to $10^{-11}\,\mathrm{A}$. (If $10^{-11}\,\mathrm{A}$ is not large enough, please turn the CURRENT RANGES Switch to $10^{-10}\,\mathrm{A}$.)
Push in the SIGNAL button to the “in” position for CALIBRATION.
Adjust the CURRENT RANGES knob until the ammeter shows that the current is zero.
Press the SIGNAL button so it moves to the “out” position for MEASURE.

Measurements

Gently pull the aperture dial away from the Photodiode Enclosure and rotate the dial so that the 2 mm aperture is aligned with the white line. Then rotate the filter wheel until the 436 nm filter is aligned with the white line. Finally remove the cover cap.
Uncover the window of the Mercury Light Source. Spectral lines of 436 nm wavelength will shine on the cathode in the phototube.
Adjust the $-4.5\,\mathrm{V}$– $30\,\mathrm{V}$ VOLTAGE ADJUST knob until the current on the ammeter is zero. Record the voltage and current.
Increase the voltage by a small amount. Record the new voltage and current.
Continue to increase the voltage by the same small increment. Record the new voltage and current each time. Stop when you reach the end of the VOLTAGE range.
Repeat the measurements with the 4mm and 8mm apertures.

Measuring the dependence on the frequency

Initial configuration

Cover the window of the Mercury Light Source enclosure with the Mercury Lamp Cap. Cover the window of the Photodiode enclosure with the Photodiode Cap.
Adjust the distance between the Mercury Light Source enclosure and Photodiode enclosure so that the general spacing is between 30.0 cm to 40.0 cm. NOTE: The recommended distance is 35.0 cm.
On the Mercury Lamp Power Supply, press the button to turn on MERCURY LAMP. On the Tunable DC (Constant Voltage) Power Supply and DC Current Amplifier, push in the POWER button to the ON position.
Allow the light source and the apparatus to warm up for 10 minutes.
On the DC (Constant Voltage) Power Supply, set the Voltage Range switch to $-4.5\,\mathrm{V}$ – $30\,\mathrm{V}$. On the DC Current Amplifier, turn the CURRENT RANGES switch to $10^{-11}\,\mathrm{A}$. (If $10^{-11}\,\mathrm{A}$ is not large enough, please turn the CURRENT RANGES Switch to $10^{-10}\,\mathrm{A}$.)
Push in the SIGNAL button to the “in” position for CALIBRATION.
Adjust the CURRENT RANGES knob until the ammeter shows that the current is zero.
Press the SIGNAL button so it moves to the “out” position for MEASURE.

Measurements

Gently pull the aperture dial and rotate it so that the 4 mm aperture is aligned with the white line. Then rotate the filter wheel until the 365 nm filter is aligned with the white line. Finally remove the cover cap.
Uncover the window of the Mercury Light Source Enclosure. Spectral lines of 365 nm will shine on the cathode in the Photodiode Enclosure.
Adjust the $-4.5\,\mathrm{V}$–$30\,\mathrm{V}$ VOLTAGE ADJUST knob so that the current display is zero. Record the voltage and current.
Increase the voltage by a small amount. Record the new voltage and current.
Continue to increase the voltage by the same small increment. Record the new voltage and current each time. Stop when you reach the end of the VOLTAGE range.
Repeat the measurements for the filter of 405 nm, 436nm and 546 nm.

Results and analysis

Planck constant

The results of the measurement are shown below (note the error bars are within the size of the dots!)

Stopping potential as a function of the frequency

The lines can be fitted with a straight line with the following parameters

Aperture size	Slope $( \mathrm{V}\,\mathrm{s} \times 10^{-15})$	Intercept $(\mathrm{V})$
2 mm	(4.12 $\pm$ 0.28)	-(1.48 $\pm$ 0.16)
4 mm	(4.17 $\pm$ 0.10)	-(1.56 $\pm$ 0.06)
8 mm	(4.08 $\pm$ 0.09)	-(1.51 $\pm$ 0.06)

From these values we can obtain the average value of Planck constant and the work function $W_0$ as \begin{align} h & = (4.12 \pm 0.31) \times 10^{-15}\,\mathrm{eV}\,\mathrm{s}\,,\\ W_0 & = (1.52 \pm 0.18)\,\mathrm{eV}\,. \end{align} The error of our meaured Planck constant with respect to its accepted value ($h=4.135\times 10^{-15}\,\mathrm{eV}\,\mathrm{s}$) is 0.28%.

Current vs Potential for constant frequency

The results of the measurement are shown in the figure below.

Current vs Potential for a fixed frequency and various intensities. The solid lines show the fitting obtained below.

We are interested to find the work function of the material and verify that it doesn't depend on the intensity of the light. Thus, we are looking for the value of the potential at which the current is no longer zero. For this, we make a linear fit of the measurements for the small values of the potential, where the figure can be appoximated by a straight line, namely we find the line $I = a+bV$ that best fits the data.

Aperture size	a $(\times 10^{-11}\,\mathrm{A})$	b $(\times 10^{-11}\,\mathrm{A}/\mathrm{V})$	$W_0\,(\mathrm{V})$
2 mm	(4.48 $\pm$ 0.22)	(2.85 $\pm$ 0.08)	(1.57 $\pm$ 0.09)
4 mm	(19.68 $\pm$ 1.35)	(12.80 $\pm$ 0.41)	(1.53 $\pm$ 0.12)
8 mm	(85.11 $\pm$ 6.36)	(54.36 $\pm$ 1.74)	(1.57 $\pm$ 0.13)

Thus, the results show that the potential at which the electrons are detached from the surface of the material is independent from the intensity of the light shone onto it. Additionally, they the work function obtained through this method is consistent with the result obtained in the previous section.

Current vs Potential for constant intensity

The results of the measurement are shown in the figure below.

Current vs Potential for a fixed intensity and various frequencies. The solid lines show the fitting obtained below.

We are interested to find the stopping potential (or, equivalently, the work function) of the material and verify that it depends on the frequency of the light. Thus, as in the previous section, we are looking for the value of the potential at which the current is no longer zero. For this, we make a linear fit of the measurements for the small values of the potential, where the figure can be appoximated by a straight line, namely we find the line $I = a+bV$ that best fits the data.

Wavelength	a $(\times 10^{-11}\,\mathrm{A})$	b $(\times 10^{-11}\,\mathrm{A}/\mathrm{V})$	Stopping potential $(\mathrm{V})$
365 nm	(2.17 $\pm$ 0.12)	(1.17 $\pm$ 0.03)	(1.85 $\pm$ 0.12)
405 nm	(9.77 $\pm$ 0.69)	(6.42 $\pm$ 0.22)	(1.52 $\pm$ 0.12)
436 nm	(18.11 $\pm$ 1.10)	(14.33 $\pm$ 0.49)	(1.26 $\pm$ 0.09)
546 nm	(12.62 $\pm$ 0.68)	(16.95 $\pm$ 0.78)	(0.74 $\pm$ 0.05)

These results confirm that the stopping potential of the electrons depends on the frequency of the light that impinges onto the surface, and the values that we have obtained through this measurement are consistent with the ones obtained in the section where we investigated the value of Planck constant.

Useful links

The data with which I made the analysis is here.
The manual to set up the experiment is here.