Particles of Light

1. The Photoelectric Effect

• The photoelectric effect was first observed in the 1880s
• The photoelectric effect is the phenomenon where electrons are emitted from the surface of a material (usually a metal) when it is exposed to light of a certain frequency.
• Figure Photoelectric effect

Key Concepts

1.1  Light as Particles

• Light behaves not only as waves but also as particles called photons.
• Each photon has a specific amount of energy determined by its frequency.

1.2  Energy of photons

The energy of a photon can be calculated using the equation: 

                      E = hf

• E is the energy of the photon,
• h is Planck’s constant (6.63×10⁻³⁴Js),
• f is the frequency of the electromagnetic wave

1.3  Work Function

• The work function (ϕ) is the minimum energy needed to remove an electron from the surface of the material.
• Different materials have different work functions.

1.4  Practical Applications

• The photoelectric effect is the principle behind devices like solar panels and photoelectric sensors.

2. Photoelectric Experiments

• There are many more experiments that can be done using different metals and a wide range of frequencies of incident light.
• It is also possible to measure the kinetic energies of the photoelectrons after they have been emitted from the surface of the metal.
• Here is a summary of these experimental observations

2.1  Threshold frequency

• It is the minimum frequency of light required to cause the photoelectric effect.
• The threshold frequency varies for different materials.
• Visible light removes photoelectrons from alkali metals, and calcium and barium.
• Other metals require ultraviolet radiation, which has a higher frequency than visible light.

2.2  Intensity

• It is rate of energy transfer per unit area.
• More photoelectrons are emitted as the intensity of light increases, but only if the frequency of the light used is above the threshold value.

 2.3  Photoelectron

• It is an electron emitted during the photoelectric effect.
• Their maximum kinetic energy depends on the frequency of the incident light so long as the frequency of this light is above the threshold frequency.

3. A Model to Explain the Photoelectric Effect

To explain threshold energy, we can describe electrons in a metal as being trapped in a potential well.
Threshold energy is the minimum amount of energy required to cause the photoelectric effect.
Potential well is a model to help us understand why electrons require energy to remove them from a metal. An electron has its least potential energy in the potential well.

• The energy needed to escape from the potential well is different for different metals, so the threshold energy varies for these metals.
• Electrons at the surface require less energy to escape compared to electrons deeper within the metal, which may be bound more strongly to atoms.
• The threshold energy is the minimum energy required to remove an electron from the metal.
• Planck’s formula for the energy of a photon, E = hf, explains the threshold frequency. Below the threshold frequency, photons do not carry enough energy to release a photoelectron so the photoelectric effect is not observed

4. Einstein’s Photoelectric Equation

Einstein’s photoelectric equation describes the relationship between the energy of incident photons and the kinetic energy of the emitted electrons. This equation is fundamental in explaining the photoelectric effect.

• The work function of a material is the least energy needed to release a photoelectron from a material. The work function has the symbol ϕ.
• Applying the conservation of energy when energy, hf, is transferred from the photon to a photoelectron:

hf = ϕ + mv²_max

• This equation is called Einstein’s photoelectric equation.
• Electrons emitted from the material’s surface have the maximum kinetic energy because energy is not used moving to the surface.
• Electrons from deeper in the material have less kinetic energy as some energy is used moving to the surface.
• Figure: Potassium’s work function is 2.29eV. Surplus energy (above the work function) becomes the kinetic energy of emitted electrons.

Examples

1. Calculate the minimum frequency of radiation that causes the photoelectric effect for aluminum. The work function for aluminum is 4.08 electron volts.
2. Calculate the maximum kinetic energy (in electron volts) of a photoelectron if light of wavelength 240nm shines on the surface.

Solution:

1. [math] E = 4.08 ev \\
   h= 6.63 × 10^{-34} Js \\
   e= 1.6× 10^{-19}C \\
   f= ? [/math]
   Convert this energy in electron volts to joules:
   [math] E = 4.08eV × 1.6 × 10^{-19}C = 6.53 × 10^{-19} J \\
   E = hf \\
   f = \frac{E}{h}= \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{240 \times 10^{-19}} = 9.8 \times 10^{14} Hz [/math]
2. Calculate the energy of a photon if its wavelength is 240nm:
   [math] E = hf \\
   E = \frac{hc}{\lambda} = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{240 \times 10^{-19}} \\
   E = 8.28 \times 10^8 \, \text{J} [/math]
   Convert joules to electron volts:
   [math] E = \frac{8.28 \times 10^8}{1.6 \times 10^{-19}} = 5.18 \, \text{eV} [/math]
   Using Einstein’s photoelectric equation, [math] KE_{max} = E − ϕ [/math] Subtract the work function from energy carried by the photon to find [math] KE_{max}[/math] [math] KE_{max} = 5.18eV − 4.0h [/math]to show how stopping potential varies with frequency.
   [math] KE_{max} = 5.18eV − 4.08eV = 1.10eV [/math]

4.1  Using the work function

• A semiconductor is a material with conductivity between a metal and an insulator
• When a material’s work function is less than 1eV, visible light can release electrons. For example, the work function for calcium is 2.87eV, potassium is 2.29eV, and cesium is 1.95eV.
• When a material’s work function is less than 77eV, infrared light can release electrons. Semiconductors are materials that have been treated so they have a low work function and respond to visible light and infrared radiation.
• Semiconductors are found in many applications, for example in CCD devices in digital cameras, and in photovoltaic cells.

4.2  Stopping potential

• When light strikes the metal surface on the left photoelectrons are emitted for some wavelengths of light.
• The photoelectrons that are emitted can be detected by a sensitive ammeter when they reach the electrode on the right of the tube.
• By making the potential of the right-hand electrode negative with respect to the left-hand electrode, the photoelectrons can be turned back, so that they do not reach the right-hand electrode.
• At this point, the current is zero, and we say that we have applied a stopping potential to the electrons.
• The stopping potential gives a measure of the electrons’ kinetic energy, because the work done by the electric field, eV, is equal to the photoelectron’s kinetic energy.
• So the electron’s kinetic energy may be calculated from the equation:
• [math] \frac{1}{2} mv^2 = eV_{\text{stop}} [/math]
• 

• Figure: Photoelectric cell

Examples

Graph to show how stopping potential varies with frequency.

1. Explain why the stopping potential changes with frequency.
2. Calculate the work function for this material.
3. Calculate the maximum kinetic energy of photoelectrons when the incident radiation is of frequency 2.5 × 1015Hz.

Solution:

1. Higher frequency photons transfer more energy to each photoelectron. The surplus energy above the work function is changed to kinetic energy of the photoelectron
2. The work function equals [math] hf [/math] when the line intercepts the x-axis.
   [math] hf = 6.63 \times 10^{-34} J s^{-1} \times 1 \times 10{15} Hz \\
   hf= 6.63 \times 10^{-19} J[/math]
3. Using the graph, the stopping potential is 6V when the incident radiation is of frequency [math] 2.5 \times 10^{15}[/math] Using [math]KE_{max} \\
   \frac{1}{2} mv^2 = eV_{\text{stop}} = e \times 6V = 6 \, \text{eV} \, \text{or} \, 9.6 \times 10^{-19} \, \text{J} [/math]

5. Electron Diffraction

Electron diffraction is a technique that allows the determination of the crystal structure of materials. When the electron beam is projected onto a specimen, its crystal lattice acts as a diffraction grating, scattering the electrons in a predictable manner, and resulting in a diffraction pattern.

5.1  The wavelength of Particles
If particles behave as waves, they must have a wavelength.
In 1923, Louis de Broglie suggested how to calculate the wavelength of particles using the particle’s momentum. The momentum of a photon, p= mc
Substituting p = mc in the equation

[math] E = mc^2 \\
E = pc \\
\text{or} \\ p = \frac{E}{c} \\
\text{Since} \\ E = hf \\
p = \frac{hf}{c} = \frac{h}{\lambda} [/math]

• De Broglie proposed that this relationship would hold for an electron, or any particle.
• The de Broglie wavelength for a particle is the Planck constant divided by a particle’s momentum; it represents the wavelength of a moving particle, or λ = h p
• He calculated that the de Broglie wavelength of electrons travelling about 0.1 to 1% of the speed of light is similar to the wavelength of X-rays.
• 

6. Diffraction using Crystals

• Diffraction is greatest when the wavelength of the wave is roughly equal to the size of the gap it passes through.
• The wavelength of X-rays is of the same order of magnitude as the spacing between ions in many crystals.
• Diffraction images were first produced using X-rays travelling through crystals in 1912.
• Since the wavelength of electrons is similar to the wavelength of X-rays, the same crystals should diffract X-rays and electrons.
• In 1927, electron diffraction was seen in experiments by Davisson and Germer, then repeated in other laboratories around the world.

Figure: Electron diffraction

6.1  Calculating the de Broglie wavelength

The de Broglie wavelength is the wavelength of a moving particle. The wavelength is calculated as Planck’s constant divided by the momentum of the particle using this equation:

[math] λ = \frac{h}{mv} [/math]

• where λ is the de Broglie wavelength in metres h is Planck’s constant m is the mass of the particle in kg, v is the velocity of the particle in [math] ms^{-1}[/math] .
• This equation shows that faster moving particles have a shorter wavelength.

This equation can also be written as:

[math] λ = \frac{h}{p} [/math]

where p is the momentum of the particle, mass × velocity.
Example
Calculate the wavelength of an electron moving at 0.1% of the speed of light. The mass of the electron is [math] 9.1 \times 10^{−31}kg [/math].

Solution:

[math] \lambda = \frac{h}{mv} \\
\lambda = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \times 3 \times 10^5} \\
\lambda = 2.4 \times 10^{-9} \, \text{m}
[/math]

7. Wave – Particle Duality

• Wave – particle duality is the idea that matter and radiation can be described best by sometimes using a wave model and sometimes using a particle model.
• Electron diffraction proves that electrons do diffract and interfere, just as waves do.
• The photoelectric effect can only be explained using a particle model of electromagnetic radiation.
• It was no longer possible to use the same model to explain the behaviour of electrons in an electric field and the behaviour of electrons when they diffract.
• The idea of wave–particle duality and the quantum theory were radically different to conventional thinking of the time.
• The quantum theory completely changed our understanding of physics and led us to a better understanding of the microscopic world.
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