Unit 21: Photons

Introduction

In the early 20th century, experiments on the interaction of light with matter revealed phenomena that could not be reconciled with the classical wave theory of light. The most striking of these was the photoelectric effect, where electrons are emitted from a metal surface when illuminated by light of sufficient frequency. This chapter explores the quantum interpretation of radiation introduced by Max Planck and expanded by Albert Einstein, leading to the modern concept of the photon.

1. Quantum Nature of Radiation

1.1 Light as Photons

According to the quantum hypothesis, electromagnetic radiation is not a continuous wave but consists of discrete packets of energy called photons. Each photon carries a quantized amount of energy that depends solely on the frequency (or wavelength) of the radiation.

1.2 Energy of a Photon

The energy E of a single photon is given by Planck’s relation:

E = h f = \frac{h c}{\lambda}

where:

• h is Planck’s constant (6.63 × 10-34 J·s),
• f is the frequency of the radiation (Hz),
• c is the speed of light in vacuum (3.00 × 108 m/s),
• λ is the wavelength (m).

Example: Calculate the energy of a photon with wavelength 500 nm.

E = \frac{h c}{\lambda} = \frac{(6.63 × 10^{-34})(3.00 × 10^{8})}{500 × 10^{-9}} ≈ 3.98 × 10^{-19} J (≈ 2.48 eV).

1.3 Failure of Wave Theory to Explain the Photoelectric Effect

Classical wave theory predicts that:

1. The kinetic energy of emitted electrons should increase with the intensity of the incident light.
2. There should be no threshold frequency; even low‑frequency light should eventually eject electrons if the intensity is high enough.
3. There should be a time delay between illumination and electron emission as energy accumulates.

Experimental observations contradict all three predictions:

• The kinetic energy of emitted electrons depends on the frequency, not the intensity, of the light.
• Below a certain frequency (the threshold frequency) no electrons are emitted, regardless of intensity.
• Emission occurs almost instantaneously (within 10^{-9} s) after illumination.

These discrepancies led Einstein (1905) to propose that light energy is quantized, and each photon transfers its entire energy to a single electron.

2. Einstein's Photoelectric Equation

2.1 Derivation

When a photon of energy hf strikes a metal surface, an electron inside the metal must overcome the attractive potential of the metal lattice before it can escape. The minimum energy required to liberate an electron is called the work function (ϕ), characteristic of the material.

Applying energy conservation:

Energy of incident photon = Work function + Kinetic energy of emitted electron

h f = ϕ + K_{max}

Re‑arranging gives Einstein’s photoelectric equation:

K_{max} = h f - ϕ

where K_{max} is the maximum kinetic energy of the photoelectrons.

2.2 Stopping Potential

To measure K_{max} experimentally, a retarding potential (V_0) is applied between the photo‑emitter and a collector. When the potential is just sufficient to stop the most energetic electrons, the electric potential energy gained by an electron equals its initial kinetic energy:

e V_0 = K_{max}

Combining with Einstein’s equation yields:

e V_0 = h f - ϕ

Thus a plot of stopping potential V_0 versus frequency f is a straight line whose slope is h/e and whose intercept on the voltage axis gives -ϕ/e.

2.3 Threshold Frequency

The threshold frequency f_0 is the minimum frequency at which photoelectric emission just begins (K_{max} = 0). Setting K_{max}=0 in Einstein’s equation gives:

h f_0 = ϕ \quad \Rightarrow \quad f_0 = \frac{ϕ}{h}

For frequencies f < f_0, no electrons are emitted irrespective of the light intensity.

3. Measurement of Planck's Constant via the Photoelectric Effect

3.1 Experimental Principle

Planck’s constant can be determined from the slope of the V_0 vs. f graph. By measuring the stopping potential for several monochromatic light sources of known frequencies, a linear fit yields:

slope = \frac{h}{e}

Multiplying the slope by the elementary charge e = 1.602 × 10^{-19} C gives h.

3.2 Experimental Setup

A typical photoelectric experiment consists of:

• Photo‑emissive cathode made of a metal (e.g., potassium, sodium, or cesium) with a well‑known work function.
• Anode kept at a variable potential relative to the cathode.
• Monochromatic light source (e.g., mercury vapour lamp with filters) providing known frequencies.
• Microammeter or electrometer to measure the photocurrent.
• Potentiometer to apply and vary the stopping potential.

The cathode is illuminated through a quartz window (transparent to UV). The anode collects emitted electrons; when the anode is made sufficiently negative relative to the cathode, the current drops to zero. The voltage at which this occurs is recorded as the stopping potential V_0 for that frequency.

3.3 Procedure

1. Set up the apparatus in a dark room to avoid stray light.
2. Select a monochromatic line (e.g., λ = 365 nm, corresponding to f ≈ 8.22 × 10¹⁴ Hz).
3. Illuminate the cathode and adjust the anode potential until the photocurrent just becomes zero; record V_0.
4. Repeat for at least four different frequencies (e.g., using filters for 405 nm, 436 nm, 546 nm, 578 nm).
5. Plot V_0 (volts) on the y‑axis against f (hertz) on the x‑axis.
6. Determine the slope of the best‑fit straight line.
7. Calculate Planck’s constant: h = slope × e.

3.4 Sample Data and Calculation

Suppose the following stopping potentials were measured:

Wavelength λ (nm)	Frequency f (×1014 Hz)	Stopping Potential V₀ (V)
365	8.22	1.25
405	7.41	0.95
436	6.88	0.78
546	5.49	0.30
578	5.19	0.15

Performing a linear regression on V_0 vs. f yields a slope of approximately 4.14 × 10^{-15} V·s. Multiplying by the elementary charge:

h = (4.14 × 10^{-15} V·s) × (1.602 × 10^{-19} C) ≈ 6.63 × 10^{-34} J·s

This value agrees with the accepted Planck’s constant within experimental uncertainty.

3.5 Sources of Error

• Contact potentials between different metals in the circuit.
• Finite bandwidth of the filters leading to uncertainty in frequency.
• Leakage currents and stray light affecting the zero‑current determination.
• Temperature variations altering the work function of the cathode.

Careful calibration, use of a reference diode, and averaging multiple measurements help reduce these errors.

4. Worked Example Problems

Example 1: Determining Work Function

A potassium surface exhibits a stopping potential of 0.54 V when illuminated by light of wavelength 400 nm. Calculate the work function of potassium.

Solution:

1. Compute photon energy: E = hc/λ.

   E = (6.63×10⁻³⁴ J·s)(3.00×10⁸ m/s) / (400×10⁻⁹ m)
         = 4.97×10⁻¹⁹ J ≈ 3.10 eV

2. Convert stopping potential to kinetic energy: K_max = e V_0 = (1.602×10⁻¹⁹ C)(0.54 V) = 8.65×10⁻²⁰ J ≈ 0.54 eV.
3. Apply Einstein’s equation: ϕ = h f - K_max = E - K_max.

   ϕ = 3.10 eV - 0.54 eV = 2.56 eV

Thus the work function of potassium is approximately 2.6 eV.

Example 2: Finding Threshold Frequency

If the work function of a metal is 2.3 eV, what is its threshold frequency?

Solution:

1. Convert work function to joules: ϕ = 2.3 eV × 1.602×10⁻¹⁹ J/eV = 3.68×10⁻¹⁹ J.
2. Use f₀ = ϕ / h:

   f₀ = (3.68×10⁻¹⁹ J) / (6.63×10⁻³⁴ J·s)
         ≈ 5.55×10¹⁴ Hz

3. Corresponding wavelength: λ₀ = c / f₀ ≈ (3.00×10⁸) / (5.55×10¹⁴) ≈ 540 nm.

Therefore, photons with wavelength shorter than about 540 nm (i.e., higher frequency) can eject electrons from this metal.

5. Summary

This chapter has established the quantum picture of light:

• Light consists of photons whose energy is given by E = hf.
• The photoelectric effect demonstrates the particle nature of radiation and cannot be explained by wave theory.
• Einstein’s photoelectric equation K_{max} = hf - ϕ relates the maximum kinetic photoelectron energy to the photon frequency and the material’s work function.
• Measuring the stopping potential for various frequencies yields a straight line whose slope provides Planck’s constant (h ≈ 6.63×10⁻³⁴ J·s).
• Experimental determination of h reinforces the validity of the quantum hypothesis and underpins modern physics and technology (e.g., photovoltaics, photodetectors).

Understanding these concepts is essential for grasping the dual nature of electromagnetic radiation and the foundation of quantum mechanics.