INTRODUCTION

By the end of the 19th century, the **wave nature of light** was well-established through Maxwell's theory of electromagnetism and Hertz's experiments on electromagnetic waves in 1887. Simultaneously, experiments on electric discharge through gases at low pressures led to significant discoveries about the fundamental constituents of matter.

Key findings during this era included the discovery of **X-rays** by Roentgen in 1895 and the **electron** by J. J. Thomson in 1897. Experiments showed that at very low gas pressure (around 0.001 mm Hg), applying an electric field caused a discharge. A glow appeared on the glass opposite the cathode, attributed to **cathode rays** emanating from the cathode. William Crookes first identified these rays in 1870, proposing in 1879 that they were streams of fast, negatively charged particles.

J. J. Thomson experimentally confirmed this hypothesis in 1897. By using perpendicular electric and magnetic fields, he measured the speed (0.1 to 0.2 times the speed of light) and the **specific charge ($e/m$)** of cathode ray particles. The value of $e/m$ ($1.76 \times 10^{11} \text{ C/kg}$) was found to be independent of the cathode material or the gas used, suggesting the **universal nature** of these particles.

Around the same time (1887), it was observed that metals emitted negatively charged particles when illuminated by ultraviolet light or when heated. These particles had the same $e/m$ ratio as cathode rays, confirming they were identical. J. J. Thomson named these particles **electrons** in 1897, proposing them as fundamental, universal constituents of matter. His work on electrons earned him the Nobel Prize in Physics in 1906.

In 1913, R. A. Millikan's oil-drop experiment precisely measured the electron's charge, finding it to be an integral multiple of $1.602 \times 10^{-19}$ C. This established the **quantisation of electric charge**. Using the measured values of $e$ and $e/m$, the electron's mass ($m$) could be determined.

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ELECTRON EMISSION

Metals contain **free electrons** responsible for their conductivity. These electrons are held within the metal surface by the attractive forces of positive ions, preventing them from escaping under normal conditions. To escape, an electron needs to gain a certain minimum amount of energy to overcome this attractive pull.

This minimum energy required for an electron to escape from the metal surface is called the **work function** ($\phi_0$) of the metal. It is typically measured in **electron volts (eV)**.

An electron volt (eV) is the energy gained by an electron when accelerated through a potential difference of 1 volt. $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$. This unit is common in atomic and nuclear physics.

The work function $\phi_0$ varies depending on the specific metal and its surface properties. For example, platinum has a high work function ($\phi_0 = 5.65 \text{ eV}$), while caesium has a low work function ($\phi_0 = 2.14 \text{ eV}$).

The energy needed for electron emission can be supplied through several physical processes:

• Thermionic emission: Heating the metal provides sufficient thermal energy to the free electrons for them to escape.
• Field emission: Applying a very strong external electric field ($\sim 10^8 \text{ V/m}$) can pull electrons out of the metal surface.
• Photoelectric emission: Illuminating the metal surface with light of suitable frequency causes electrons to be emitted. These emitted electrons are called **photoelectrons**.

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PHOTOELECTRIC EFFECT

The emission of electrons from a metal surface when light falls on it is known as the **photoelectric effect**. This phenomenon was first observed by Heinrich Hertz in 1887.

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Hertz’S Observations

During his electromagnetic wave experiments, Hertz observed that sparks produced were stronger when the emitter plate was illuminated by ultraviolet light. This suggested that light was somehow assisting the release of charged particles from the metal surface. Electrons near the surface absorb energy from the incident light, enabling them to overcome the binding forces and escape.

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Hallwachs’ And Lenard’S Observations

Wilhelm Hallwachs and Philipp Lenard conducted more detailed investigations (1886-1902). Lenard observed that current flowed in a vacuum tube when ultraviolet radiation struck the emitter plate and stopped when the radiation ceased (Figure 11.1). This confirmed that light caused the emission of electrons, which formed a current.

Hallwachs' experiments with a charged zinc plate showed that UV light caused it to lose negative charge or gain positive charge, further indicating the emission of negative particles (electrons). After the electron's discovery, these were identified as photoelectrons.

A crucial observation by Hallwachs and Lenard was that photoemission only occurs if the incident light's frequency is greater than a certain minimum value, called the **threshold frequency** ($\nu_0$). This frequency is characteristic of the metal; below $\nu_0$, no emission happens, regardless of light intensity. Metals like zinc require UV, while alkali metals are sensitive to visible light.

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EXPERIMENTAL STUDY OF PHOTOELECTRIC EFFECT

A standard experimental setup (Figure 11.1) is used to study the photoelectric effect, involving a photosensitive emitter (C), a collector (A), and an ac source with variable voltage/polarity, illuminated by monochromatic light. Photocurrent is measured with a microammeter.

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Effect Of Intensity Of Light On Photocurrent

With fixed frequency and potential, increasing the intensity of incident light leads to a **linear increase in photoelectric current** (Figure 11.2). This means the number of photoelectrons emitted per second is directly proportional to the intensity of incident radiation.

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Effect Of Potential On Photoelectric Current

With fixed frequency and intensity, varying the potential difference between the plates:

• As the collector A is made more positive (accelerating potential), the photocurrent increases until it reaches a maximum value called **saturation current**, where all emitted photoelectrons are collected.
• As A is made negative (retarding potential), the photocurrent decreases. At a specific minimum negative potential, the **stopping potential ($V_0$)**, the photocurrent becomes zero. This voltage stops even the most energetic photoelectrons. The maximum kinetic energy ($K_{max}$) of emitted electrons is related to $V_0$ by $\mathbf{K_{max} = eV_0}$ (Equation 11.1).

Figure 11.3 shows that increasing light intensity increases saturation current but **does not change the stopping potential** for a given frequency. $K_{max}$ is independent of intensity.

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Effect Of Frequency Of Incident Radiation On Stopping Potential

With fixed intensity, varying the frequency ($\nu$) of incident light:

• Different frequencies result in different stopping potentials (Figure 11.4). Higher frequencies require a more negative $V_0$, indicating higher $K_{max}$ for photoelectrons.

Plotting $V_0$ versus $\nu$ yields a **straight line** for each metal (Figure 11.5). The slope is constant, but the intercept on the frequency axis varies, representing the **threshold frequency ($\nu_0$)** below which $V_0$ is zero (no emission). $K_{max}$ depends linearly on $\nu$ but not intensity.

Summary of experimental observations:

• Photocurrent $\propto$ intensity (above $\nu_0$).
• $V_0$ (and $K_{max}$) depends on $\nu$ and material, not intensity.
• Threshold frequency $\nu_0$ exists; no emission below it.
• Photoemission is practically **instantaneous** ($\sim 10^{-9}$ s), even for low intensity, if $\nu > \nu_0$.

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PHOTOELECTRIC EFFECT AND WAVE THEORY OF LIGHT

The classical wave theory described light as a continuous electromagnetic wave. It predicted that energy absorption by electrons would be continuous and distributed over the wavefront.

Based on this wave picture, the following would be expected for photoelectric emission:

• $K_{max}$ of photoelectrons should increase with light intensity (higher amplitude means more energy absorbed). **Contradicts observation.**
• A threshold frequency should not exist; any frequency, given enough time and intensity, should supply enough energy. **Contradicts observation.**
• Energy absorption by individual electrons would take time, especially at low intensity. Emission should not be instantaneous. **Contradicts observation.**

The wave picture of light fundamentally failed to explain the key features of the photoelectric effect.

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EINSTEIN’S PHOTOELECTRIC EQUATION: ENERGY QUANTUM OF RADIATION

In 1905, Albert Einstein proposed a revolutionary picture of electromagnetic radiation to explain the photoelectric effect. He suggested that light energy is not continuous but consists of discrete packets or bundles of energy called **quanta** (later called **photons**).

Each photon has energy $\mathbf{E = h\nu}$, where $h$ is Planck's constant and $\nu$ is the light frequency. Photoelectric emission occurs when an electron absorbs a single photon. If the photon energy ($h\nu$) exceeds the minimum energy needed for the electron to escape the metal ($\phi_0$, work function), the electron is emitted with maximum kinetic energy.

$\mathbf{K_{max} = h\nu - \phi_0}$ (Equation 11.2)

This is **Einstein's photoelectric equation**, based on energy conservation. Energy of photon = Work function + Maximum kinetic energy of emitted electron.

This picture successfully explains the experimental observations:

• $K_{max}$ depends linearly on $\nu$ and is independent of intensity: The equation directly shows this. Emission results from a single photon absorption, so its energy ($h\nu$) determines $K_{max}$, not the number of photons (intensity).
• Threshold frequency ($\nu_0$) exists: For emission, $K_{max} \ge 0$, so $h\nu \ge \phi_0$. This implies a minimum frequency $\mathbf{\nu_0 = \phi_0/h}$ (Equation 11.3) is required.
• Photocurrent $\propto$ intensity: Higher intensity means more photons per second, leading to more electrons absorbing photons and being emitted, thus increasing photocurrent (for $\nu > \nu_0$).
• Instantaneous emission: The elementary process of photon absorption is instantaneous. If $h\nu \ge \phi_0$, emission happens immediately. Low intensity means fewer events, not delayed events.

Using $K_{max} = eV_0$ (Equation 11.1), the equation can be written as $\mathbf{eV_0 = h\nu - \phi_0}$, or $\mathbf{V_0 = \frac{h}{e}\nu - \frac{\phi_0}{e}}$ (Equation 11.4). This predicts a linear $V_0$ vs $\nu$ graph with slope $h/e$, independent of the material. Millikan's experiments (1906-1916) confirmed this linearity and determined $h$, matching Planck's value, solidifying the acceptance of the photon concept.

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WAVE NATURE OF MATTER

The previous discussion highlights the **dual nature of radiation** (light), exhibiting both wave-like (interference, diffraction, polarisation) and particle-like (photoelectric effect, Compton effect) properties depending on the phenomenon observed. The wave picture is useful for propagation (like focusing by a lens), while the particle picture is needed for interaction/absorption (like by retinal cells).

A natural question arises: If radiation (energy) has this dual nature, does **matter** (particles like electrons, protons, etc.) also exhibit wave-like properties? In 1924, French physicist Louis Victor de Broglie proposed the bold hypothesis that moving material particles should also possess wave characteristics under suitable conditions, based on the symmetry between matter and energy.

De Broglie proposed that the wavelength ($\lambda$) associated with a moving particle is related to its momentum ($p$) by:**

$\mathbf{\lambda = \frac{h}{p} = \frac{h}{mv}}$ (Equation 11.5)

where $h$ is Planck's constant, $m$ is the particle's mass, and $v$ is its speed. This is the **de Broglie relation**, and the associated wave is called a **matter wave** or **de Broglie wave**.

The de Broglie relation shows the duality: $\lambda$ (wave property) is related to $p=mv$ (particle property) by $h$. This relation also holds for photons ($\lambda_{photon} = c/\nu = h/(h\nu/c) = h/p_{photon}$).

From the formula, wavelength $\lambda$ is smaller for more massive particles (large $m$) or faster particles (large $v$). For macroscopic objects in everyday life (large $m$, significant $v$), $p=mv$ is very large, leading to an extremely small de Broglie wavelength ($\sim 10^{-34}$ m for a ball). This makes the wave properties of macroscopic objects unobservable in practice.

However, for sub-atomic particles like electrons (small $m$), even at relatively low speeds, the de Broglie wavelength can be significant and measurable, comparable to atomic dimensions. For an electron accelerated from rest through a potential $V$, its kinetic energy $K = eV$, momentum $p = \sqrt{2mK} = \sqrt{2meV}$. Its de Broglie wavelength is $\lambda = h/\sqrt{2meV}$. Substituting numerical values, this simplifies to $\mathbf{\lambda \approx \frac{1.227}{\sqrt{V}} \text{ nm}}$ when $V$ is in volts and $\lambda$ in nanometers (Equation 11.11). For $V=120$ V, $\lambda \approx 0.112$ nm, comparable to atomic spacing in crystals. This suggested that electron wave nature could be verified by crystal diffraction experiments.

De Broglie received the Nobel Prize in Physics in 1929 for his hypothesis.

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Photocell

A **photocell** is a device that utilises the photoelectric effect to convert changes in light intensity into changes in electric current. It is sometimes called an "electric eye".

It typically consists of a photosensitive metal plate (cathode) and a collector (anode) inside an evacuated glass or quartz bulb. When light of suitable wavelength (frequency above the threshold) falls on the photosensitive cathode, photoelectrons are emitted. These electrons are attracted to the anode by an applied voltage, resulting in a measurable photocurrent.

Photocells have numerous applications:

• Light detection and measurement (e.g., in cameras).
• Control systems (e.g., automatic door openers, counting devices, burglar/fire alarms that respond to interrupted light beams).
• Sound reproduction in motion pictures.
• Scanning in television cameras.
• Detecting flaws in industrial production lines.

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Probability Interpretation To Matter Waves

Understanding the precise physical meaning of a matter wave is challenging. Max Born provided a crucial interpretation: the **intensity** (square of the amplitude, $|A|^2$) of the matter wave at a point in space is proportional to the **probability density** of finding the particle at that point. Probability density means probability per unit volume.

So, $|A|^2 \Delta V$ is the probability of finding the particle in a small volume $\Delta V$ around that point. Where the wave intensity is high, the probability of finding the particle is high.

This interpretation connects the wave picture (amplitude) to the particle picture (probability of location). For a particle with a definite momentum, its de Broglie wave has a single wavelength and extends over all space. Born's rule implies the particle is not localized, meaning its position uncertainty is infinite, consistent with Heisenberg's uncertainty principle ($\Delta x \Delta p \approx \hbar$), where zero momentum uncertainty implies infinite position uncertainty.

More realistically, a particle is described by a **wave packet**, a superposition of waves localised in space ($\Delta x$ finite). A wave packet inherently has a spread of wavelengths, leading to a spread of momenta ($\Delta p$) via the de Broglie relation, consistent with the uncertainty principle ($\Delta x \Delta p \approx \hbar$).

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DAVISSON AND GERMER EXPERIMENT

The wave nature of electrons, predicted by de Broglie, was experimentally verified by C.J. Davisson and L.H. Germer in 1927, and independently by G.P. Thomson in 1928, through **electron diffraction** experiments using crystals. Davisson and Thomson shared the Nobel Prize in 1937 for this discovery.

The Davisson-Germer experiment used an **electron gun** to produce a collimated beam of electrons accelerated to a controlled velocity by a variable voltage. This electron beam was directed to strike a nickel crystal. Electrons were scattered by the crystal's atoms in various directions.

An **electron detector** measured the intensity of the scattered electron beam at different scattering angles. The apparatus was in an evacuated chamber.

Davisson and Germer measured the scattered electron intensity for accelerating voltages from 44 V to 68 V. They observed a strong peak in intensity at an accelerating voltage of **54 V** for a scattering angle of **50°**. This peak indicated constructive interference of the electron waves scattered from the ordered arrangement of atoms in the nickel crystal, similar to how X-rays are diffracted by crystals.

From the diffraction pattern, they calculated the wavelength of the electron waves to be **0.165 nm**.

They then calculated the de Broglie wavelength for an electron accelerated through 54 V using the formula $\lambda = 1.227/\sqrt{V}$ nm:

$\lambda = \frac{1.227}{\sqrt{54}} \text{ nm} \approx 0.167 \text{ nm}$.

The excellent agreement between the experimentally determined wavelength (0.165 nm) and the theoretically predicted de Broglie wavelength (0.167 nm) provided strong confirmation of de Broglie's hypothesis and the wave nature of electrons.

Subsequent experiments, like electron double-slit experiments and interference with molecules, further confirmed the wave nature of matter. The de Broglie hypothesis is a cornerstone of quantum mechanics and has led to applications like the electron microscope, which uses the wave properties of electrons for high-resolution imaging.

SUMMARY

This chapter explores the **dual nature of radiation (light)** and proposes the **wave nature of matter**. It starts with the photoelectric effect, which provides strong evidence for the particle nature of light.

• **Work function ($\phi_0$)** is the minimum energy required for an electron to escape a metal surface. Emission can be by heat (thermionic), electric field (field emission), or light (photoelectric).
• **Photoelectric effect** is electron emission from metals illuminated by light of suitable frequency. Key features: photocurrent $\propto$ intensity (above $\nu_0$), $K_{max}$ (or stopping potential $V_0$) depends on $\nu$ and material but not intensity, threshold frequency $\nu_0$ exists (no emission below it), emission is instantaneous. $K_{max} = eV_0$.
• Classical wave theory failed to explain photoelectric effect features.
• **Einstein's photoelectric equation**: $K_{max} = h\nu - \phi_0$. Light energy is in discrete **photons** ($E=h\nu$). One photon-one electron interaction. Explains all observations. $V_0 = (h/e)\nu - \phi_0/e$. Millikan's experiments verified this and $h$.
• **Particle nature of light (photon):** Photon has energy $h\nu$ and momentum $p=h\nu/c = h/\lambda$. Confirmed by Compton effect.
• **Wave nature of matter:** De Broglie hypothesis: Moving particles also have wave properties. **De Broglie wavelength** $\lambda = h/p = h/(mv)$. Duality in $\lambda$ and $p$. Measurable for sub-atomic particles, negligible for macroscopic objects. $\lambda_{electron} = 1.227/\sqrt{V}$ nm.
• **Davisson-Germer experiment** (and G.P. Thomson): Experimental verification of electron wave nature through crystal diffraction, matching de Broglie's prediction.
• Matter waves incorporate Heisenberg's uncertainty principle and support Bohr's quantisation condition.
• **Photocell:** Technological application of photoelectric effect, converting light intensity to current changes.

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POINTS TO PONDER

Reflections on concepts:

• Free electrons in metals are confined within the surface; $\phi_0$ is the minimum escape energy.
• Electron energies in metals have a distribution; $\phi_0$ corresponds to the least energy needed by the most energetic electrons.
• Photoelectric observations imply energy absorption in discrete units $h\nu$, suggesting quantisation of light energy, not necessarily that light *is* particles at all times and places.
• Stopping potential observations ($V_0$ vs intensity and frequency) are crucial evidence distinguishing wave vs photon pictures.
• De Broglie wavelength $\lambda$ is physically significant; phase velocity $v_p = \nu\lambda$ is not. Group velocity equals particle velocity.

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Table of relevant physical quantities:

Physical Quantity	Symbol	Dimensions	Unit	Remarks
Planck’s constant	$h$	[ML$^2$T$^{–1}$]	J s	$E = h\nu$
Stopping potential	$V_0$	[ML$^2$T$^{–3}$A$^{–1}$]	V	$eV_0 = K_{max}$
Work function	$\phi_0$	[ML$^2$T$^{–2}$]	J; eV	$K_{max} = h\nu – \phi_0$
Threshold frequency	$\nu_0$	[T$^{–1}$]	Hz	$\nu_0 = \phi_0/h$
de Broglie wavelength	$\lambda$	[L]	m	$\lambda = h/p$

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ADDITIONAL EXERCISES

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