Photons

AS UNIT 2

Electricity and light

2.7 Photons   

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) The fact that light can be shown to consist of discrete packets (photons) of energy
b) How the photoelectric effect can be demonstrated
c) How a vacuum photocell can be used to measure the maximum kinetic energy, [math]E_{k \,  max}[/math], of emitted electrons in eV and hence in J
d) The graph of [math]E_{k \,  max}[/math] against frequency of illuminating radiation
e) How a photon picture of light leads to Einstein’s equation, [math]E_{k \,  max} = hf – Φ[/math], and how this equation correlates with the graph of [math]E_{k \,  max}[/math] against frequency
f) The fact that the visible spectrum runs approximately from 700 nm (red end) to 400 nm (violet end) and the orders of magnitude of the wavelengths of the other named regions of the electromagnetic spectrum
g) Typical photon energies for these radiations
h) How to produce line emission and line absorption spectra from atoms
i) The appearance of such spectra as seen in a diffraction grating
j) Simple atomic energy level diagrams, together with the photon hypothesis, line emission and line absorption spectra
k) How to determine ionization energies from an energy level diagram
l) The demonstration of electron diffraction and that particles have a wave-like aspect
m) The use of the relationship [math]p = \frac{h}{λ}[/math]  for both particles of matter and photons
n) The calculation of radiation pressure on a surface absorbing or reflecting photons

Specified Practical Work

o   Determination of h using LEDs

  • ⇒ The Photon Nature of Light and the Photoelectric Effect

  • a)    Light as Discrete Packets of Energy (Photons)

  • Light exhibits both wave-like and particle-like properties, which is known as wave-particle duality. In some experiments, light behaves as a wave (e.g., diffraction and interference), while in others, it behaves as a stream of discrete energy packets called photons
  • According to Planck’s quantum theory, the energy (E) of each photon is quantized and given by:
  • [math]E = hf[/math]
  • Where:
  • – h = Planck’s constant ( [math]6.626 × 10^{-34} Js[/math]),
  • – f = frequency of the light.
  • This means that light of a higher frequency (e.g., ultraviolet) consists of photons with more energy than light of a lower frequency (e.g., red light).
  • Figure 1 Photon emission

  • b)   Demonstration of the Photoelectric Effect

  • The photoelectric effect is the emission of electrons from a metal surface when it is illuminated by light of a sufficiently high frequency.
  • ⇒  Experimental Setup for Demonstrating the Photoelectric Effect
  • A clean metal plate (e.g., zinc or sodium) is placed inside a vacuum photocell.
  • A monochromatic light source (such as a UV lamp) is directed at the metal surface.
  • A variable power supply is connected to the photocell to measure the stopping potential (voltage needed to halt the emitted electrons).
  • A sensitive ammeter (microammeter) is used to detect the photocurrent (flow of emitted electrons).
  • Figure 2 Photoelectric effect
  • ⇒  Observations
  • 1. If the light frequency is too low, no electrons are emitted, regardless of intensity.
  • – This contradicts classical wave theory, which predicts that increasing the intensity of light should eventually eject electrons.
  • 2. If the light frequency is above a threshold value, electrons are emitted instantly.
  • – The energy of the photons depends on frequency, not intensity.
  • 3. Increasing light intensity increases the number of emitted electrons, but not their maximum kinetic energy ([math]E_{k max}[/math]).
  • – More photons mean more electrons are ejected, but their energy remains the same.
  • 4. Increasing the light frequency increases the maximum kinetic energy of the emitted electrons.

  • c)    The Role of the Vacuum Photocell in Measuring ([math]E_{k max}[/math]

  • A vacuum photocell can be used to measure the maximum kinetic energy ( [math]E_{k max}[/math]) of the emitted electrons.
  • ⇒  How It Works:
    1. When photons strike the metal cathode, electrons are ejected due to the photoelectric effect.
    2. Some of these electrons reach the anode, creating a measurable photoelectric current.
    3. A variable voltage is applied in the opposite direction (opposing the electron flow).
    4. As the voltage increases, fewer electrons reach the anode, and the current decreases.
    5. At a certain voltage (called the stopping potential ​[math]V_s[/math]), the current drops to zero because even the most energetic electrons cannot overcome the opposing voltage.
  • Since the maximum kinetic energy of the ejected electrons is converted into electrical potential energy, we use the equation:
  • [math]E_{k max} = eV_s[/math]
  • Where:
  • – e = charge of an electron ([math]1.602 × 10^{-19} C[/math] ),
  • – [math]V_s[/math] = stopping potential.
  • This allows us to determine [math]E_{k max}[/math]​ in electron-volts (eV) and then convert it to joules (J) using:
  • [math]1eV = 1.602 × 10^{-19} J[/math]

  • d)   The Graph of  ​[math]E_{k max}[/math] Against Frequency

  • ⇒   Experimental Results
  • By plotting the maximum kinetic energy of emitted electrons ( [math]E_{k max}[/math]​) against the frequency of the incident light, we get a straight-line graph with the following characteristics:
  • [math]E_{k max} = hf – ϕ[/math]
  • ⇒  Key Features of the Graph:
    1. Slope = h (Planck’s constant), showing that the energy of photons increases linearly with frequency.
    2. x-intercept = [math]f_o[/math] (Threshold frequency)
  • Figure 3 Graph between [math]E_{k max}[/math] and frequency
  • – Below this frequency, no electrons are emitted.
  • 3. [math] \text{y-intercept} = -ϕ[/math] (Work function of the metal)
  • – The minimum energy required to eject an electron from the metal surface.
  • ⇒   Einstein’s Photoelectric Equation
  • [math]E_{k max} = hf – ϕ[/math]
  • Where:
  • – [math]E_{k max}[/math] = maximum kinetic energy of the emitted electrons,
  • – h = Planck’s constant,
  • – f = frequency of the incident light,
  • – ϕ = work function of the metal (minimum energy needed to release an electron).
  • This is the maximum kinetic energy of the ejected electrons. If we look at the graph above, we can see that this match. The threshold frequency [math]f_T[/math] is when the maximum kinetic energy is zero. Here
  • [math]0 = h f_T – \phi \\
    f_T = \frac{\phi}{h}[/math]
  • ⇒  Explanation of the Equation
  • – A photon with energy hf is absorbed by an electron in the metal.
  • – Part of this energy (ϕ) is used to free the electron from the metal surface.
  • – Any remaining energy is converted into kinetic energy of the emitted electron.
  • If the photon energy is less than the work function (ϕ), no electrons are emitted.

  • e)    Importance of the Photon Model of Light

  • The photoelectric effect provided conclusive evidence that light behaves as particles (photons) rather than just waves, which could not be explained by classical physics.
  • – Wave theory prediction: Electrons should be emitted for all frequencies if intensity is high enough.
  • – Photon theory prediction (confirmed by experiments): No emission occurs below a threshold frequency, regardless of intensity.
  • This led to the development of quantum mechanics, where energy is quantized in discrete packets (quanta), fundamentally changing our understanding of light and matter interactions.
  • Figure 4 Electrons emission the metal surface
  • ⇒ Conclusion
  • – Light consists of photons, each carrying energy [math]E = hf[/math]
  • – The photoelectric effect demonstrates that electrons are emitted from a metal surface only when light of a sufficiently high frequency (above threshold [math]f_o[/math]) is incident.
  • The vacuum photocell allows measurement of electron energy and stopping potential to confirm Einstein’s equation.
  • The graph of [math]E_{k max}[/math]​ vs. frequency supports Einstein’s photon model and gives the work function of the metal.
  • The experiment confirmed that light exhibits particle-like behavior, leading to the development of quantum mechanics.
  • This experiment was one of the key pieces of evidence supporting quantum physics, earning Einstein the Nobel Prize in Physics in 1921.

  • e)    The Visible Spectrum and the Electromagnetic Spectrum

  • The visible spectrum refers to the range of electromagnetic (EM) waves that the human eye can detect. It runs approximately from 700 nm (red light) to 400 nm (violet light).
  • Beyond the visible spectrum, there are other regions of the electromagnetic spectrum, each with different wavelengths and photon energies:
Region Wavelength Range Approximate Photon Energy (eV)
Radio waves >1m <  [math]10^{-6}[/math]eV
Microwaves 1mm−1m [math]10^{-6} – 10^{-3}[/math]
Infrared (IR) 700nm−1mm   [math]10^{-3} – 1[/math]eV
Visible light 400nm−700nm 1.7−3.1 eV
Ultraviolet (UV) 10nm−400nm 3.1−124 eV
X-rays 0.01nm−10nm [math]124−10^{4124} – 10^{4124}−104[/math] eV
Gamma rays <0.01nm >[math]110^4[/math]eV

  • Figure 5 Visible light- The electromagnetic spectrum
  • The photon energy for a given wavelength is calculated using:
  • [math]E = \frac{hc}{λ}[/math]
  • Where:
  • – E = photon energy (J),
  • – h = [math]6.626 × 10^{-34} Js[/math](Planck’s constant),
  • – c = [math]3.00 × 10^8 m/s[/math](speed of light),
  • – λ = wavelength (m).

  • h)   Production of Line Emission and Line Absorption Spectra

  • Atoms have discrete energy levels for their electrons. When an electron transitions between energy levels, it either absorbs or emits a photon of energy:
  • [math]E_{\text{photon}} = h f = E_{\text{higher}} – E_{\text{lower}}[/math]
  • Where:
  • – [math]E_{\text{photon}}[/math]= energy of emitted or absorbed photon,
  • – [math]E_{\text{lower}}[/math]= energy of the atomic states
  • – f = frequency of the photon.
  • a)      Emission Spectra
  • When an electron drops from a higher to a lower energy level, it emits a photon of a specific frequency.
  • If the emitted photons pass through a diffraction grating, they form a line emission spectrum, unique for each element.
  • Example: The Balmer series in hydrogen produces visible spectral lines.
  • b) Absorption Spectra
  • When white light passes through a cool gas, some photons are absorbed by electrons in atoms, causing excitation.
  • These absorbed photons have specific energies corresponding to electronic transitions.
  • When viewed through a diffraction grating, the continuous spectrum (from white light) has dark absorption lines at specific wavelengths.
  • Figure 6  Emission and absorbance of Line spectra
  • Difference:
  • – Emission spectrum = bright lines on a dark background.
  • – Absorption spectrum = dark lines on a continuous spectrum background.

  • i)   Appearance of Spectra Through a Diffraction Grating

  • A diffraction grating consists of many closely spaced slits that cause interference patterns of light.
  • – When an atomic emission spectrum is observed using a diffraction grating, sharp spectral lines appear, each corresponding to a specific wavelength of light emitted by the atom.
  • – The angle at which the spectral lines appear depends on the diffraction equation:
  • [math]nλ = dsinθ[/math]
  • Where:
  • – n = diffraction order (1st, 2nd, etc.),
  • – λ = wavelength of light,
  • – d = spacing between adjacent slits in the grating.
  • – θ = diffraction angle.
  • For emission spectra, bright lines appear at characteristic angles.
  • For absorption spectra, dark lines appear against a continuous background.

  • j)     Atomic Energy Level Diagrams and the Photon Hypothesis

  • Atoms have discrete energy levels, represented as horizontal lines in an energy level diagram.
  • – Electrons can move between levels by absorbing or emitting
  • – The lowest energy level is the ground state.
  • – Higher levels are excited states.
  • – The difference in energy levels corresponds to the energy of the emitted or absorbed photon.
  • Example: Hydrogen Energy Levels
Energy Level Energy (eV)
n = ∞ 0.00 eV
n = 3 -1.51 eV
n = 2 -3.40 eV
n = 1 -13.60 eV
  • Figure 7 Model of hydrogen spectra
  • – For a transition from n = 3 to n = 2, the photon energy is:
  • [math]E = (-3.40) – (-1.51) = 1.89 eV[/math]
  • – This corresponds to a wavelength in the visible spectrum.
  • The visible spectrum extends from 700 nm (red) to 400 nm (violet), and the entire EM spectrum spans many orders of magnitude in wavelength and energy.
  • Photons are discrete packets of energy, with [math]E = hf[/math]
  • Emission spectra result from electron de-excitations, while absorption spectra result from photon absorption by atoms.
  • Diffraction gratings reveal sharp spectral lines corresponding to different wavelengths.
  • Atomic energy levels are discrete, and photon interactions occur in quantized steps.
  • The line spectra of elements provide fundamental evidence for quantized atomic energy levels and are crucial in astronomical spectroscopy and quantum mechanics.
  • This understanding forms the basis of modern quantum physics and has applications in lasers, spectroscopy, and atomic physics.

  • k)   Determining Ionization Energies from an Energy Level Diagram

  • Ionization energy is the minimum energy required to remove an electron completely from an atom in its ground state.
  • Figure 8 Determine the ionization energy level diagram
  • ⇒ Using an Energy Level Diagram
  • – The highest energy level (n = ∞) represents a free electron with zero energy (by convention).
  • – The ground state (n = 1) is the lowest energy level of the atom and has the most negative energy value.
  • – The ionization energy is the energy required to move an electron from the ground state (n = 1) to n = ∞.
  • – It is given by:
  • [math]E_{ionization} = E_∞ – E_1[/math]
  • Since [math]E_∞ = 0[/math], the ionization energy is simply: [math]E_{ionization}[/math] For example, in hydrogen, the ground state energy is -13.6 eV, so the ionization energy is +13.6 eV.

  • l) Demonstration of Electron Diffraction and Wave-Particle Duality

  • ⇒ Electron Diffraction Experiment
  • This experiment provides evidence that particles exhibit wave-like behavior.
  • A beam of electrons is accelerated through a potential difference and directed toward a thin graphite or nickel foil.
  • The crystal lattice of the foil acts as a diffraction grating, causing the electrons to form an interference pattern on a fluorescent screen.
  • This circular pattern of rings is similar to the diffraction pattern of light waves, proving wave-particle duality.
  • Figure 9 Electron Diffraction
  • ⇒ Wave-Particle Duality
  • Wave-like behavior: Electrons undergo diffraction and interference, behaviors typically associated with waves.
  • Particle-like behavior: When electrons are detected, they appear as individual particles at specific points.
  • de Broglie Hypothesis: Particles, like electrons, have an associated wavelength, given by:
  • [math]λ = \frac{h}{p}[/math]
  • Where:
  • – λ = wavelength (m),
  • – h = Planck’s constant ( [math]6.63 × 10^{-34} Js[/math]),
  • – p = momentum ( [math]mv[/math]), where mmm is mass and v is velocity.

  • m)   The Relationship ​[math]p = \frac{h}{λ}[/math] for Particles and Photons

  • This equation is fundamental to quantum mechanics and describes the wave nature of particles:
  • –     For Photons:
  • Photons have no rest mass, but they have momentum given by:
  • [math]p_{photon} = \frac{h}{λ}[/math]
  • This equation is derived from the energy-momentum relationship and Planck’s equation [math]E = hf[/math]
  • ⇒  For Matter Particles (de Broglie Wavelength):
  • – Matter particles also exhibit wave-like behavior, with a de Broglie wavelength:
  • [math] λ = \frac{h}{mv} [/math]
  • Where mmm is the mass and v is the velocity of the particle.
  • – This explains electron diffraction, where electrons behave like waves with a characteristic wavelength.

  • n)   Radiation Pressure on a Surface Absorbing or Reflecting Photons

  • Radiation pressure is the force per unit area exerted by photons when they strike a surface.
  • ⇒   For a Surface that Absorbs Photons:
  • – Each photon carries momentum:
  • [math]p = \frac{h}{λ}[/math]
  • The radiation pressure is given by:
  • [math]P = \frac{\text{power}}{\text{area} \times c}[/math]
  • where c is the speed of light.
  • ⇒  For a Surface that Reflects Photons:
  • – If the surface reflects photons, the momentum change is doubled, so the pressure is:
  • [math]P = \frac{2 \times \text{power}}{\text{area} \times c}[/math]
  • ⇒  Applications of Radiation Pressure:
  • – Solar sails in space propulsion use radiation pressure from the Sun.
  • – Laser trapping of microscopic particles relies on radiation pressure.

  • Specified Practical Work

  • Determination of h using LEDs

  • ⇒ Objective:
  • To determine the value of Planck’s constant (h) by measuring the threshold voltage (V) of different Light Emitting Diodes (LEDs) and using the equation:
  • [math]E = hf[/math]
  • Where:
  • – E is the energy of a photon emitted by the LED,
  • – h is Planck’s constant (to be determined),
  • – f is the frequency of the emitted light.
  • ⇒  Theory:
  • When an LED is forward biased, electrons gain energy from an external voltage source and move from the conduction band to the valence band of the semiconductor material.
  • – When they recombine with holes, they release energy in the form of photons (light).
  • – The minimum energy required to emit light corresponds to the threshold voltage (V), where the LED just begins to glow.
  • – The energy of the emitted photon is:
  • [math]E = eV[/math]
  • Where:
  • – e is the charge of an electron ( [math]1.6 × 10^{-19} C[/math]),
  • – V is the threshold voltage of the LED.
  • Since the photon energy is also given by:
  • [math]E = hf[/math]
  • equating both equations:
  • [math]hf = eV[/math]
  • Rearranging for Planck’s constant:
  • [math]h = \frac{eV}{f}[/math]
  • Figure 10 Determine of h using LED
  • ⇒  Apparatus:
    1. A set of LEDs of different colors (wavelengths)
    2. A variable power supply or battery pack
    3. A digital voltmeter to measure V
    4. A resistor to limit current through the LED
    5. A spectrometer or a table of LED wavelengths
    6. Connecting wires
  • ⇒  Experimental Procedure:
  • 1. Set up the circuit
  • – Connect the LED in series with a resistor and a variable power supply.
  • – Connect a voltmeter across the LED to measure the voltage.
  • 2. Increase the voltage slowly
  • – Adjust the power supply until the LED just begins to emit light.
  • – This voltage is the threshold voltage V.
  • 3. Repeat for different LEDs
  • – Use LEDs of different colors (red, green, blue, etc.) with known wavelengths λ.
  • – Record the threshold voltage V for each LED.
  • 4. Determine the frequency
  • – The frequency of emitted light is:
  • [math]f = λc[/math]
  • Where:
  • – [math]c = 3.00 × 10^8 m/s[/math](speed of light),
  • – λ is the LED’s wavelength in meters.
  • 5. Plot a graph of V f
  • – The equation [math]eV = hf[/math] suggests a linear relationship between V and f.
  • – The gradient of the straight-line graph will be:
  • [math]h = \frac{\text{gradient}}{e}[/math]
  • Since[math]V = \frac{hf}{e}[/math] ​, the gradient of the graph is [math]\frac{h}{e}[/math] .
  • ⇒  Data Analysis and Calculations:
  • Plot a graph of V (y-axis) vs. f (x-axis).
  • – The slope of the straight-line is [math][/math].
  • – Multiply by the electron charge [math]e = 1.6 × 10^{-19} C[/math] to get h:
  • [math]h = (\text{slope}) × e[/math]
  • The accepted value of Planck’s constant is:
  • [math]h = 6.63 × 10^{-34} J⋅s[/math]
  • Compare your experimental value with the accepted value.
  • ⇒  Example Calculations:
  • For an LED emitting red light (λ=650 nm, V=8 V):
  • 1. Find frequency:
  • [math]f = \frac{c}{\lambda} \\
    f = \frac{3 \times 10^8}{650 \times 10^{-9}} \\
    f = 4.62 \times 10^{14} \text{ Hz}[/math]
  • 2. Calculate Planck’s constant:
  • [math]h = \frac{eV}{f} \\
    h = \frac{(1.6 \times 10^{-19}) \times (1.8)}{4.62 \times 10^{14}} \\
    h = 6.23 \times 10^{-34} \text{ J·s}[/math]
  • This is close to the accepted value.
  • ⇒  Sources of Error:
  • – Uncertainty in voltage measurement (threshold voltage may not be exact).
  • – LEDs may not emit monochromatic light (wavelength may vary slightly).
  • – Temperature effects on LED performance.
  • – Spectral measurement inaccuracies (using a spectrometer is more precise).
  • ⇒  Conclusion:
  • This experiment provides a practical method for determining Planck’s constant.
  • The method demonstrates the quantum nature of light and electron energy levels.
  • The experiment supports the photoelectric equation and confirms the relationship [math]E = hf[/math] 
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