The Planck constant measurement experiment examines the fact that light energy is exchanged not continuously, but in discrete units.
The energy of a single photon is proportional to the frequency of light, and the proportionality constant is the Planck constant. The Planck constant is a fundamental physical constant in quantum mechanics and is used to explain the photoelectric effect, atomic spectra, semiconductors, blackbody radiation, and other phenomena.
In school and university laboratories, a common method is to measure the light-emission threshold voltage and emission wavelength of light-emitting diodes (LEDs), then compare the electrical energy lost by electrons with the energy of the emitted photons.
This article assumes an experiment in which the light-emission threshold voltages of LEDs of several colors are measured and the Planck constant is determined from their relationship with frequency. It is organized in the order of introduction, safety notes, items to record, sample experimental values, calculations, sources of error, example results, and discussion examples.
Note:
This article is a study reference for discussing results obtained in physics experiments at schools, universities, and similar institutions. The experimental conditions and measured values shown here are examples for explanation and are not actual measurement results.
LEDs have polarity and may be damaged by excessive current. Always connect an appropriate resistor and check the limits of the supply voltage and current. Do not look directly at high-brightness or ultraviolet LEDs. Follow your institution’s laboratory manual and the instructions of your instructor or TA during actual experiments.
- What Is the Planck Constant?
- Measurement Principle Using LEDs
- How to Determine the Light-Emission Threshold Voltage
- Main Items to Record in the Results
- Sample Experimental Conditions
- Sample Experimental Values
- Calculation Methods
- Main Sources of Error
- Example of How to Write the Results
- Points to Connect to the Discussion
- Discussion Example
- Summary
What Is the Planck Constant?
The Planck constant is the proportionality constant connecting photon energy and frequency.
E = hν
Here, E is the energy of one photon, h is the Planck constant, and ν is the frequency of light.
The defined value of the Planck constant is:
h = 6.62607015 × 10−34 J·s
In the current International System of Units, the Planck constant is defined to have exactly this value.
Measurement Principle Using LEDs
When a forward voltage is applied to an LED, electrons and holes recombine in the semiconductor and emit light corresponding to their energy difference.
The electrical energy gained by one electron through a voltage V is:
E = eV
Here, e is the elementary charge.
Ideally, the electrical energy of an electron at the LED light-emission threshold voltage V0 is considered to correspond to the energy of the emitted photon.
eV0 ≈ hν
The frequency of light is calculated from wavelength as follows.
ν = c/λ
Combining these relations gives:
V0 ≈ (h/e)ν
If light-emission threshold voltage is plotted on the vertical axis and frequency on the horizontal axis, an ideal result is a straight line. If its slope is a, the Planck constant is:
h = ea
How to Determine the Light-Emission Threshold Voltage
An LED does not suddenly begin emitting fully at one exact voltage. A very small current flows even at low voltage, and emission intensity increases continuously.
Therefore, several methods can be used to determine the threshold voltage.
- The voltage at which emission is visually detected in a dark room
- The voltage when current reaches a specified value
- The intercept obtained by extrapolating the linear portion of the current-voltage characteristic
- The voltage at which a light-sensor output exceeds the background level
The visual method is simple, but it is affected by the observer’s visual sensitivity and differences in sensitivity to LED color. For greater accuracy, extrapolation of the current-voltage characteristic or use of a light sensor is preferable.
Main Items to Record in the Results
- LED emission color
- LED model number
- Center or peak wavelength
- Method used to measure wavelength
- Frequency
- Light-emission threshold voltage
- Current at the emission threshold
- Series resistance
- Supply voltage
- Current-voltage characteristic
- Graph of threshold voltage versus frequency
- Slope of the regression line
- Intercept of the regression line
- Planck constant
- Relative error from the reference value
- Voltmeter resolution
- Room temperature
- Method used to determine the threshold voltage
Sample Experimental Conditions
| Item | Sample condition |
|---|---|
| LEDs used | Red, orange, yellow, green, blue |
| Series resistor | 330 Ω |
| Power supply | Variable DC power supply |
| Voltage measurement range | 0–5 V |
| Voltmeter resolution | 0.01 V |
| Ammeter resolution | 0.01 mA |
| Threshold determination | Extrapolation of the linear part of the I-V characteristic |
| Wavelength | LED specification value or spectrometer measurement |
| Room temperature | 23 ℃ |
Sample Experimental Values
LED Wavelength, Frequency, and Light-Emission Threshold Voltage
| Emission color | Wavelength λ (nm) | Frequency ν (1014 Hz) | Threshold voltage V0 (V) |
|---|---|---|---|
| Red | 635 | 4.72 | 1.54 |
| Orange | 605 | 4.96 | 1.64 |
| Yellow | 590 | 5.08 | 1.69 |
| Green | 525 | 5.71 | 1.95 |
| Blue | 470 | 6.38 | 2.23 |
Example Current-Voltage Characteristic of a Red LED
| LED voltage (V) | Current (mA) |
|---|---|
| 1.30 | 0.01 |
| 1.40 | 0.03 |
| 1.50 | 0.10 |
| 1.60 | 0.45 |
| 1.70 | 1.20 |
| 1.80 | 2.45 |
| 1.90 | 4.10 |
Example Repeated Measurements
| LED | 1st (V) | 2nd (V) | 3rd (V) | Average (V) |
|---|---|---|---|---|
| Red | 1.53 | 1.54 | 1.55 | 1.54 |
| Orange | 1.63 | 1.64 | 1.65 | 1.64 |
| Yellow | 1.68 | 1.69 | 1.70 | 1.69 |
| Green | 1.94 | 1.95 | 1.96 | 1.95 |
| Blue | 2.22 | 2.23 | 2.24 | 2.23 |
Calculation Methods
Converting Wavelength to Frequency
For a red LED with wavelength 635 nm and the speed of light 3.00 × 108 m/s:
ν = c/λ
= (3.00 × 108)/(635 × 10−9)
= 4.72 × 1014 Hz
Energy per Electron Supplied to the LED
If the threshold voltage of the red LED is 1.54 V:
E = eV0
= (1.602 × 10−19) × 1.54
= 2.47 × 10−19 J
Photon Energy
Using the reference value of the Planck constant, the photon energy at 635 nm is:
E = hc/λ
= (6.626 × 10−34) × (3.00 × 108)/(635 × 10−9)
= 3.13 × 10−19 J
The electrical energy calculated from the LED threshold voltage does not exactly match the photon energy because of internal voltage drops, nonradiative recombination, and other effects in the LED.
Determining the Planck Constant from the Slope of the Regression Line
Suppose a linear fit to threshold voltage versus frequency gives a slope of 4.10 × 10−15 V·s.
V0 = aν + b
a = h/e
h = ea
= (1.602 × 10−19) × (4.10 × 10−15)
= 6.57 × 10−34 J·s
Finding the Slope from Two Points
If only the red and blue LED values are used:
a = (V2 − V1)/(ν2 − ν1)
= (2.23 − 1.54)/{(6.38 − 4.72) × 1014}
= 4.16 × 10−15 V·s
h = ea
= (1.602 × 10−19) × (4.16 × 10−15)
= 6.66 × 10−34 J·s
However, using only two points makes the result strongly affected by errors in each measurement, so a least-squares regression using several LEDs is more appropriate.
Relative Error
For an experimental value of 6.57 × 10−34 J·s and a reference value of 6.626 × 10−34 J·s:
Relative error (%) = |experimental value − reference value|/reference value × 100
= |6.57 − 6.626|/6.626 × 100
= 0.85%
Meaning of the Intercept
For actual LEDs, the threshold-voltage-versus-frequency graph may not pass through the origin and may have an intercept.
V0 = (h/e)ν + b
The intercept may reflect junction potential differences, electrode contact potentials, constant biases caused by the measurement method, nonradiative processes, and related effects.
Main Sources of Error
| Source of error | Effect on measurement | Likely trend in results | Improvement |
|---|---|---|---|
| Visual judgment of emission onset | Judgment differs among observers | Larger scatter in voltage values | Use extrapolation of the I-V characteristic |
| Color dependence of human visual sensitivity | Blue and red may appear dimmer | Threshold criterion varies by color | Use a light sensor |
| LED wavelength width | Emission has a distribution rather than one wavelength | Frequency depends on representative wavelength chosen | Measure peak wavelength with a spectrometer |
| Difference between specified and actual wavelength | Frequency on the horizontal axis shifts | Regression slope changes | Measure the actual emission spectrum |
| LED-to-LED variation | Forward voltage differs even for the same color | Points deviate from the line | Measure several LEDs and average |
| Temperature rise | Forward voltage decreases | Voltage becomes smaller after prolonged operation | Use low current and measure quickly |
| Voltmeter resolution | Threshold voltage cannot be read finely | Greater slope uncertainty | Use a higher-resolution voltmeter |
| Voltmeter zero offset | Adds a constant bias to all voltages | Mainly shifts the regression intercept | Calibrate before measurement |
| Contact resistance | Voltage drops occur in wiring and terminals | LED terminal voltage is measured incorrectly | Measure directly across the LED terminals |
| Resistor tolerance | Current and I-V calculations shift | Extrapolated threshold changes | Use the measured resistance value |
| Nonradiative recombination | Not all electrical energy becomes light | eV and hν do not simply coincide | Evaluate from the regression slope |
| Voltage drops outside the junction | Includes voltage not directly contributing to emission | Creates an intercept in the graph | Determine slope using multiple-point regression |
| Too few measurement points | One error strongly affects the slope | Poor reproducibility of the Planck constant | Use more LED colors |
| Narrow frequency range | Slope is difficult to determine accurately | Relative error increases | Use a broad range from red to blue |
Why Higher-Frequency LEDs Have Higher Threshold Voltages
A higher-frequency photon has a larger energy hν. Blue light has a higher frequency than red light, so a blue LED requires electron-hole recombination across a larger energy difference.
Therefore, the light-emission threshold voltage of a blue LED is generally higher than that of a red LED.
Why Threshold Voltage and Frequency Have a Linear Relationship
If the electrical energy gained by one electron in an LED is eV and the emitted photon energy is hν, the two are approximately proportional.
Thus, the threshold voltage is approximately a linear function of frequency, and its slope corresponds to h/e.
Why the Graph Does Not Pass Through the Origin
In an actual LED, not all applied voltage is converted into photon energy. Voltage drops outside the semiconductor junction, contact potentials, nonradiative recombination, and the definition of the threshold voltage all contribute, so the regression line can have a nonzero intercept.
Why Visual Observation Produces Color-Dependent Error
Human eyes have different sensitivities at different wavelengths. In general, yellow-green light is perceived more readily, while red or blue may appear dim even at the same radiant intensity.
Therefore, when emission onset is judged visually, the criterion changes with LED color and can introduce systematic error into the threshold voltage.
Example of How to Write the Results
For five LEDs—red, orange, yellow, green, and blue—the light-emission threshold voltage was determined from the current-voltage characteristic.
The threshold voltage increased as the emission wavelength became shorter and the frequency became higher. A graph of threshold voltage on the vertical axis and frequency on the horizontal axis showed an approximately linear relationship.
The slope of the regression line was 4.10 × 10−15 V·s, and the Planck constant calculated using the elementary charge was 6.57 × 10−34 J·s.
The relative error from the reference value of 6.626 × 10−34 J·s was 0.85%.
Points to Connect to the Discussion
- Did the threshold voltage increase with frequency?
- Was the graph of threshold voltage versus frequency linear?
- Was the Planck constant obtained from the regression slope close to the reference value?
- Was the graph intercept zero?
- How was the threshold voltage determined?
- Was there color-dependent visual sensitivity in visual judgment?
- Was the actual peak wavelength of each LED used?
- Could the LED emission spectral width be neglected?
- Was there any temperature rise during measurement?
- Was the voltage measured directly across the LED terminals?
- Was there variation among individual LEDs?
- Were the number of measurement points and frequency range sufficient?
- Was a two-point calculation or regression line more appropriate?
Discussion Example
In this experiment, the light-emission threshold voltage and emission wavelength were measured for LEDs of several colors, and the Planck constant was determined from the relationship between threshold voltage and light frequency.
The measurements showed that the threshold voltage increased as the wavelength shortened from red to blue. The threshold voltage was 1.54 V for the red LED and 2.23 V for the blue LED.
Because the energy of one photon is expressed as E = hν, higher-frequency light has greater energy. In an LED, the energy difference associated with electron-hole recombination is emitted as a photon, so producing higher-frequency blue light requires more electrical energy than producing red light.
The graph of threshold voltage versus frequency was approximately linear. Its regression slope was 4.10 × 10−15 V·s, and the Planck constant calculated using h = ea was 6.57 × 10−34 J·s.
This value was close to the reference value of 6.626 × 10−34 J·s, with a relative error of 0.85%. Therefore, the Planck constant could be approximately determined from the relationship between LED threshold voltage and frequency.
However, the regression line did not pass exactly through the origin. If the ideal relation eV = hν applied directly, it would pass through the origin, but in a real LED not all applied voltage is converted into photon energy.
Inside an LED, voltage drops due to resistance outside the junction, contact potential differences, energy transfer to lattice vibrations, and nonradiative recombination occur. These effects can produce a nonzero intercept in the relationship between threshold voltage and frequency.
The method used to determine the threshold voltage is another source of error. If emission onset is judged visually, human visual sensitivity depends on wavelength, so yellow-green LEDs may be visible at low intensity while red and blue LEDs may remain difficult to see even when they are emitting.
As a result, the threshold criterion may differ by color, affecting the linearity and slope of the graph.
In this experiment, the threshold voltage was obtained by extrapolating the linear portion of the I-V characteristic, reducing subjective error compared with visual observation. However, the intercept can still depend on which range is selected as the linear portion.
If nominal LED wavelength values are used, they may differ from the actual emission peak wavelengths. In addition, LED emission has a finite spectral width rather than a single wavelength, so the calculated frequency depends on the choice of representative wavelength.
Wavelength error changes the horizontal axis of the graph and therefore directly affects both the regression slope and the calculated Planck constant. Measuring each LED’s peak wavelength with a spectrometer would improve accuracy.
Furthermore, an LED’s forward voltage decreases as its temperature rises. If a large current is applied for a long time during measurement, the LED heats up and later voltage measurements may become smaller.
To reduce this effect, current should not be made unnecessarily large, measurements should be completed quickly, and the LED should be allowed to cool between measurements.
When the slope is determined from only two LEDs, errors in the individual measurements strongly affect the result. Using LEDs of several colors and determining the slope by least-squares regression reduces the influence of individual measurement errors.
Overall, the LED light-emission threshold voltage showed an approximately linear relationship with light frequency, and the Planck constant could be obtained from the slope. Differences from the reference value were likely caused by threshold-voltage determination, LED spectral width, differences between nominal and measured wavelengths, temperature rise, contact potential differences, and related factors.
Discussion Example When the Planck Constant Is Larger Than the Reference Value
Possible reasons for obtaining a Planck constant larger than the reference value include judging the threshold voltage of high-frequency LEDs too high, judging the low-frequency voltage too low, or estimating the blue LED wavelength as longer than its actual value and therefore calculating its frequency too low.
Discussion Example When the Planck Constant Is Smaller Than the Reference Value
Possible reasons for obtaining a Planck constant smaller than the reference value include judging the threshold voltage of high-frequency LEDs too low, a reduction in the blue LED forward voltage due to heating during measurement, or estimating the wavelength as shorter than its actual value.
Discussion Example When the Graph Is Not Linear
Possible reasons why the graph of threshold voltage versus frequency was not linear include differences in LED structure and material, color-dependent changes in the criterion for threshold determination, differences between nominal and actual peak wavelengths, individual LED variation, and temperature differences.
Discussion Example When the Regression Line Does Not Pass Through the Origin
Possible reasons why the regression line did not pass through the origin include voltage drops outside the junction, contact potential differences, nonradiative recombination, and constant bias arising from the method used to determine the light-emission threshold voltage. Therefore, it is important to determine the Planck constant from the slope of the regression line rather than from its intercept.
Summary
The Planck constant is the physical constant in the photon energy-frequency relation E = hν.
In an LED experiment, the light-emission threshold voltage is plotted against light frequency, and the Planck constant is determined from the slope.
The threshold voltage increases with light frequency, but in actual LEDs the regression line does not necessarily pass through the origin because of contact potentials, nonradiative recombination, temperature changes, and other effects.
In the discussion, it is important to connect visual judgment, spectral width, individual LED variation, temperature rise, and the regression method with how the measured value differs from the reference value.
