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All matter can be modeled as a particle or a wave. In this report, electrons, in particular, are described to have wave-particle duality. De Broglie proposed that electrons can travel as waves and can exhibit behavior like diffraction where it was found that as the speed of the accelerated electrons increased, the de Broglie wavelength of electrons decreased. Einstein proposed that light is composed of particles called photons with energy E = hv, and the photoelectric effect illustrated that increasing the photon's frequency increased the kinetic energy of the photoelectron emitted; they are directly proportional.
The photoelectric effect is the emission of electrons from a metal surface when light shines on it.
The ejected electrons are referred to as photoelectrons (e–) and their kinetic energy is proportional to the energy of the photon striking the metal. By providing evidence for electrons behaving like particles, the photoelectric effect defied the classical wave theory of light. The classical notion is that of electron diffraction in which the emission of electrons is in the form of waves.
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It provides evidence for the wave nature of electrons. In this report, the results of both the photoelectric effect and electron diffraction experiments are discussed to observe both the particle and wave-like behavior of an electron. The idea of wave-particle duality is presented with the de Broglie equation, λ = h / mu, equating to the wavelength of an electron and turning out to be true for all particles (particles have wave properties).
With the cell covered and apparatus switched off, the auto-zero button was pushed on the picometer.
The apparatus was switched on, and the lamp was placed around 10-15 cm above the photocell. The multimeter was turned on. Each filter was placed over the cell, and the voltage dial on the apparatus was adjusted until the picometer read as zero. The voltage reading on the multimeter was recorded (V0). Stop voltage for each of the six filters, respectively (2.87, 2.58, 2.31, 2.09, 1.90, 1.75).
The EHT knob was turned anti-clockwise, and the power was switched on. The voltage was increased, and the diffraction pattern became visible. The diameters of the two concentric rings were measured, using a pair of calipers, for 5 values of the accelerating voltage between 2,000 and 3,500 V (i.e., 10 measurements in all). For each measurement, the diffraction angle was calculated, followed by the 10 values of sin(θ). Voltage values in kV (2.0, 2.5, 2.75, 3.0, 3.5); outer ring diameter in m for each voltage (0.058, 0.052, 0.050, 0.048, 0.044), diffraction angle for each measurement (0.433, 0.388, 0.373, 0.358, 0.328), sin(θ1) (6.415, 5.708, 5.475, 5.661, 4.787); inner ring diameter in m for each voltage (0.034, 0.031, 0.029, 0.028, 0.026), diffraction angle for each measurement (0.254, 0.231, 0.216, 0.209, 0.194), sin(θ2) (3.679, 3.339, 3.119, 3.016, 2.797).
For each filter used, the stop voltage was recorded (V0). In addition, the wavelengths were used in the equation, v = cλ, to calculate the transmitted frequencies for each filter (v). Such results are seen in Table 1.
| Filter No | Wavelength (nm) | Stop Voltage (V0) | Frequency (s⁻¹) |
|---|---|---|---|
| 1 | 405 | 2.87 | 7.41 x 10¹⁴ |
| 2 | 450 | 2.58 | 6.67 x 10¹⁴ |
| 3 | 500 | 2.31 | 6 x 10¹⁴ |
| 4 | 550 | 2.09 | 5.45 x 10¹⁴ |
| 5 | 600 | 1.90 | 5 x 10¹⁴ |
| 6 | 650 | 1.75 | 4.62 x 10¹⁴ |
From Table 1, it is clear that as the wavelength increases, the transmitted frequency decreases. Concerning the photoelectric effect, this conveys that as the wavelength of the photon hitting the metal surface decreases, its frequency decreases, thus they are inversely proportional.
The energy of a photon is expressed by E = hv, where h is equal to the Planck constant, 6.626 x 10⁻³⁴. Planck proposed that the light hitting the metal surface could only exist in discrete quanta. Using this equation, it is found that frequency is proportional to the energy carried by the photon and that the higher the photon frequency, the more energy the photon has. This can be seen in the values from Table 1. By taking the highest frequency of 7.41 x 10¹⁴ s⁻¹ and multiplying this by the Planck constant, a value of 4.91 x 10⁻¹⁹ J is obtained. On the other hand, if the lowest frequency value is used, 4.62 x 10¹⁴ s⁻¹ x h, a value of 3.06 x 10⁻¹⁹ J is obtained. By comparing the two, it is clear that the higher the photon frequency, the higher its energy, and the lower the frequency, the lower the energy the photon has. From this, the photoelectric effect can be described as so. If a photon of high enough energy is shone on a metal, electrons will be emitted from the metal. A photon that is below a certain threshold frequency will not cause any electrons to be emitted, and a photon that is above a certain frequency will cause electrons to be ejected. Furthermore, the kinetic energy of the photoelectrons is proportional to the light frequency. So, as the photon frequency increases, the kinetic energy of the photoelectron increases. The kinetic energy of emitted electrons is measured by applying a stopping voltage, V0. The equation for the kinetic energy of a photoelectron is KEe = e V0, where e is the charge of an electron. By taking the stopping voltage, 2.87 V of the highest frequency value, and multiplying this by the electron charge, a value of 4.59 x 10⁻¹⁹ J is obtained. Whereas, the stop voltage 1.75 V of the lowest frequency leads to a value of 2.8 x 10⁻¹⁹ J when multiplied by e. Thus, this conveys when the photon's frequency increases, the kinetic energy of the emitted electrons increases, and when the photon's frequency decreases, the kinetic energy decreases too.
The photoelectric effect gives rise to emission spectra. The electrons emitted are in an excited state and can return to a lower energy level, losing energy as it does so. This energy released is in the form of a photon. The energy levels are quantized, so each of the transitions, to a lower state, produces a photon of different energy and hence, frequency. Thus, this causes the formation of the line emission spectra where only light of specific frequencies is produced.
Figure 1: A graph of stop voltage against frequency with the equation for the line of best fit to be y = 4E-15x - 0.1081. An estimate of the error in the slope is 1.93 x 10⁻¹⁷
The diameters of the outer and inner rings were measured, in addition to their diffraction angles. Such results are illustrated in Table 2.
| Measurement | Voltage (kV) | Electron Wavelength (m) | Outer Ring Diameter (xo m) | sin(4θ1) | Inner Ring Diameter (xi m) | sin(4θ2) |
|---|---|---|---|---|---|---|
| 1 | 2.0 | 8.68 x 10⁻¹⁰ | 0.058 | 0.433 | 0.034 | 0.254 |
| 2 | 2.5 | 7.77 x 10⁻¹⁰ | 0.052 | 0.388 | 0.031 | 0.231 |
| 3 | 2.75 | 7.40 x 10⁻¹⁰ | 0.050 | 0.373 | 0.029 | 0.216 |
| 4 | 3.0 | 7.09 x 10⁻¹⁰ | 0.048 | 0.358 | 0.028 | 0.209 |
| 5 | 3.5 | 6.57 x 10⁻¹⁰ | 0.044 | 0.328 | 0.026 | 0.194 |
The mean reading for d1 was 3.898 ± 0.032 nm (n = 5)
The mean reading for d2 was 6.74 ± 0.040 nm (n = 5)
It is clear from Table 2 that as the voltage is increased, the outer and inner ring diameters decrease. For example, low voltage 2.0 kV and high voltage 3.5 kV have outer ring diameters of 0.058m and 0.044m, respectively, thus portraying this trend. Moreover, the energy of an electron accelerated increases with increasing voltage. This is expressed by E = eV, where e is the charge of an electron. For example, with a high voltage like 3.5kV, E = e x 3.5 = 5.6 x 10⁻¹⁹ J. Compared to a low voltage like 2.0kV, E = e x 2.0 = 3.2 x 10⁻¹⁹ J. Therefore, voltage and the energy of an electron are directly proportional.
This shows that as the speed of the accelerated electrons increases, the smaller the de Broglie wavelength of electrons becomes, expressed as λ = h / mu. Moreover, from table 2, it is clear that the smaller the diameter of the rings, the smaller the de Broglie wavelength of electrons too.
The de Broglie wavelength of electrons is calculated using several equations: the energy of an electron accelerated by a voltage, E = eV, and the kinetic energy of an electron, E = 1/2meu². By using these, a calculated value for velocity, u, is obtained which is then used in the de Broglie equation to determine the wavelength.
The purpose of these experiments was to illustrate the concept of wave-particle duality: light, or electrons in this case, can behave both as a particle or a wave. The photoelectric effect could be thought of as light composed of particles called photons with their energy proportional to their frequency, E = hv. De Broglie suggested that electrons also hold wave properties like wavelength and suggests that as the speed of electrons increases, the de Broglie wavelength of electrons decreases, λ = h / mu.
These findings provide strong evidence for the wave-particle duality of electrons and support the quantum theory of light. Light and electrons can behave as both particles and waves, depending on the experimental conditions and the properties being observed.
In conclusion, the experiments conducted in this study contribute to our understanding of the fundamental nature of matter and light. They confirm that matter, such as electrons, exhibits wave-like and particle-like properties simultaneously, a phenomenon known as wave-particle duality. This duality challenges classical physics and is a fundamental concept in quantum mechanics, shaping our comprehension of the behavior of particles at the quantum level.
Wave-Particle Duality of Electrons and Photoelectric Effect. (2024, Jan 06). Retrieved from https://studymoose.com/document/wave-particle-duality-of-electrons-and-photoelectric-effect
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