Introduction

By the late 19th century, the understanding of light was dominated by its wave nature, strongly supported by Maxwell's electromagnetic theory and experiments by Hertz demonstrating the generation and detection of electromagnetic waves. However, concurrent investigations into the conduction of electricity through gases at low pressures within discharge tubes led to discoveries that challenged the purely wave picture.

Significant milestones included the discovery of X-rays by Roentgen in 1895 and the electron by J.J. Thomson in 1897. Experiments with discharge tubes revealed that at very low pressures, an electric discharge occurred between the electrodes. A notable observation was a fluorescent glow on the glass wall opposite the cathode, attributed to rays emanating from the cathode, termed cathode rays. William Crookes first discovered these rays and suggested they were streams of fast-moving negatively charged particles.

J.J. Thomson confirmed this particle hypothesis experimentally. By subjecting cathode rays to perpendicular electric and magnetic fields, he was able to measure their velocity and the ratio of their charge to mass ($e/m$). The measured speeds were a fraction of the speed of light, and the $e/m$ value was found to be constant regardless of the cathode material or the gas in the tube. This universality suggested that these particles were fundamental constituents of matter.

Around the same period, it was observed that metals emitted negatively charged particles when exposed to ultraviolet light or when heated to high temperatures. The $e/m$ ratio for these particles was identical to that of cathode rays. These findings established that these particles, produced through different means, were fundamentally the same. J.J. Thomson named them electrons in 1897, proposing them as fundamental, universal components of matter. His groundbreaking work led to the Nobel Prize in Physics in 1906.

Later, in 1913, R.A. Millikan's famous oil-drop experiment precisely measured the charge of an electron, demonstrating that electric charge exists in discrete multiples of a fundamental unit, approximately $1.602 \times 10^{-19}$ C. Combining Millikan's charge value with Thomson's $e/m$ ratio allowed for the determination of the electron's mass.

Electron Emission

Metals are known to have a sea of "free" electrons that contribute to their electrical conductivity. These electrons are not bound to specific atoms but move freely within the metal lattice. However, they are generally confined within the metal because the surface of the metal exerts attractive forces (from the positively charged ions) that pull the electrons back. To escape the metal surface, an electron must overcome this attractive potential barrier.

The minimum energy required for an electron to escape from the surface of a metal is called the work function ($\phi_0$) of that metal. This property varies depending on the specific metal and the condition of its surface. The work function is commonly measured in units of electron volts (eV). One electron volt is defined as the kinetic energy gained by an electron when it is accelerated through an electric potential difference of 1 volt. $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$.

There are several ways to supply this minimum energy to the electrons to facilitate their emission from the metal surface:

(i) Thermionic Emission: When a metal is heated to a sufficiently high temperature, the thermal energy transferred to the free electrons can exceed the work function, allowing them to escape the surface.

(ii) Field Emission: Applying a very strong electric field (on the order of $10^8$ V/m) to the metal surface can pull electrons out. This is used in applications like spark plugs.

(iii) Photoelectric Emission: When light of a suitable frequency illuminates a metal surface, electrons are emitted. These emitted electrons are called photoelectrons, and the phenomenon is known as the photoelectric effect.

Photoelectric Effect

Hertz’s Observations

The first observation related to the photoelectric effect was made by Heinrich Hertz in 1887. While conducting experiments on generating and detecting electromagnetic waves using spark gaps, Hertz noticed that the sparks produced in his detector loop were more vigorous when the metal electrodes were illuminated by ultraviolet light from the source spark. This suggested that the ultraviolet light was somehow enhancing the electric discharge, implying that charged particles were being released from the metal surface by the action of light.

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Hallwachs’ And Lenard’s Observations

Following Hertz's initial finding, Wilhelm Hallwachs and Philipp Lenard conducted detailed investigations into this effect. Lenard used an evacuated glass tube containing two metal electrodes. He observed that when ultraviolet radiation fell on one electrode (the emitter), a current flowed in the circuit, indicated by a galvanometer. The current stopped as soon as the radiation was removed. This confirmed that light was causing the emission of charged particles from the metal, which were then collected by the other electrode (the collector) due to the applied voltage.

Hallwachs further studied the effect using a charged zinc plate connected to an electroscope. He found that a negatively charged zinc plate lost its charge when illuminated by ultraviolet light. An uncharged zinc plate became positively charged under UV illumination, and a positively charged plate became even more positively charged. These observations strongly indicated that negatively charged particles were being emitted from the zinc surface under the influence of UV light.

After the electron was discovered in 1897, it became clear that the emitted negatively charged particles were electrons. Hallwachs and Lenard's experiments also revealed crucial characteristics of this emission: there was a minimum frequency of incident light (called the threshold frequency, $\nu_0$) below which no electrons were emitted, regardless of how intense the light was. This threshold frequency was found to be different for different metals. Metals like zinc, cadmium, and magnesium required ultraviolet light, while alkali metals like sodium and potassium were sensitive to visible light. Substances that exhibit this property are called photosensitive materials. The emitted electrons were termed photoelectrons, and the phenomenon itself was named the photoelectric effect.

Experimental Study Of Photoelectric Effect

A standard experimental setup to study the photoelectric effect involves an evacuated glass or quartz tube containing two metal electrodes: a photosensitive plate (emitter, C) and a collector plate (A) (

Figure 11.1). Light of controlled frequency and intensity is directed onto the emitter plate through a window. Electrons emitted from C are collected by A, creating a measurable electric current (photocurrent) in the external circuit. A battery provides a variable potential difference between C and A, and a commutator allows reversing the polarity so that A can be made positive (accelerating potential) or negative (retarding potential) relative to C. A voltmeter measures the potential difference, and a microammeter measures the photocurrent.

Using this apparatus, experiments were performed to study the variation of photocurrent with incident light intensity, frequency, the potential difference between electrodes, and the material of the emitter plate.

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Effect Of Intensity Of Light On Photocurrent

With the frequency of the incident light and the potential difference between C and A kept constant (A at a positive potential to collect electrons), the intensity of the light falling on C is varied. It is observed that the photocurrent is directly proportional to the intensity of the incident light (

Figure 11.2). Since photocurrent is a measure of the number of photoelectrons emitted per second, this experiment showed that the number of photoelectrons emitted is directly proportional to the intensity of the incident radiation.

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Effect Of Potential On Photoelectric Current

Keeping the frequency and intensity of the incident light fixed, the potential of the collector plate A relative to the emitter plate C is varied. Initially, with A at a positive potential, increasing the positive voltage increases the photocurrent. At a certain positive potential, the current reaches a maximum value and does not increase further even if the voltage is raised. This is called the saturation current, indicating that all the photoelectrons emitted from C are being collected by A.

If the potential of A is made negative (retarding potential) with respect to C, the photocurrent decreases. Only electrons with sufficient kinetic energy can overcome the repulsive force from A and reach it. As the negative potential is increased, fewer electrons reach A, and the current drops. Eventually, at a specific minimum negative potential, the photocurrent becomes zero. This potential is called the stopping potential ($V_0$) or cut-off potential.

At the stopping potential $V_0$, even the most energetic photoelectrons with maximum kinetic energy ($K_{max}$) are just prevented from reaching the collector. Thus, the work done by the stopping potential on these electrons is equal to their maximum kinetic energy:

$$K_{max} = eV_0$$

where $e$ is the elementary charge of the electron.

Repeating this experiment with the same frequency but different intensities reveals that while the saturation current increases with increasing intensity (more electrons emitted), the stopping potential ($V_0$) remains unchanged (

Figure 11.3). This means that for a given frequency, the maximum kinetic energy of the photoelectrons is independent of the intensity of the incident light.

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Effect Of Frequency Of Incident Radiation On Stopping Potential

Now, keeping the intensity of incident light constant, the experiment is performed with different frequencies ($\nu$) of light. It is observed that for different frequencies (above a certain value), the stopping potential ($V_0$) required to stop the photocurrent is different (

Figure 11.4). Higher frequencies require a greater negative stopping potential, indicating that the maximum kinetic energy of photoelectrons increases with the frequency of incident light. The saturation current remains the same for different frequencies at the same intensity (since the number of emitted electrons is proportional to intensity).

A graph plotting the stopping potential ($V_0$) against the frequency ($\nu$) of the incident radiation for a given photosensitive material shows a straight line (

Figure 11.5). This graph has two key implications:

(i) The stopping potential, and thus the maximum kinetic energy of photoelectrons ($K_{max} = eV_0$), varies linearly with the frequency of incident radiation.

(ii) There is a minimum frequency, the threshold frequency ($\nu_0$), below which the stopping potential is zero (meaning no emission occurs, $K_{max}=0$), regardless of the intensity. This threshold frequency is characteristic of the material and corresponds to the frequency at which photoemission just begins.

These experiments also showed that if the frequency of incident light is above the threshold frequency, photoelectric emission is almost instantaneous, with no detectable time delay (less than $10^{-9}$ s), even for very low light intensities.

In summary, the key experimental observations of the photoelectric effect are:

• Photocurrent is directly proportional to light intensity (for $\nu > \nu_0$).
• Saturation current is proportional to intensity, but stopping potential is independent of intensity (for a given $\nu > \nu_0$).
• Existence of a threshold frequency ($\nu_0$) below which no emission occurs, regardless of intensity.
• Above $\nu_0$, $K_{max}$ (or $V_0$) increases linearly with frequency, independent of intensity.
• Photoelectric emission is instantaneous (for $\nu > \nu_0$).

Photoelectric Effect And Wave Theory Of Light

The wave picture of light, which successfully explained interference, diffraction, and polarisation, faced significant challenges in explaining the observed features of the photoelectric effect. According to the classical wave theory:

1. Energy in a wave is spread out continuously over the entire wavefront. Electrons at the metal surface should absorb energy continuously from the incident radiation.

2. A higher intensity wave means larger electric and magnetic field amplitudes, implying more energy is delivered to the surface per unit area per unit time. Therefore, the energy absorbed by each electron should increase with increasing light intensity. This would lead to the expectation that the maximum kinetic energy of emitted electrons should increase with intensity.

3. With continuous energy absorption, even light of low frequency should be able to cause electron emission if it's intense enough or shines for a long enough duration to allow electrons to accumulate sufficient energy to overcome the work function.

These predictions from the wave theory directly contradict the experimental observations:

• Experiment shows $K_{max}$ is independent of intensity, not dependent.
• Experiment shows a distinct threshold frequency, below which no emission occurs even at high intensities. Wave theory predicts no such sharp threshold based on frequency alone.
• Experiment shows instantaneous emission, even at very low intensities. Wave theory, considering energy distributed over the wavefront and shared among many electrons, predicts a time delay for an individual electron to accumulate enough energy, potentially hours for very dim light.

Therefore, the classical wave theory fails to explain the fundamental characteristics of the photoelectric effect.

Einstein’s Photoelectric Equation: Energy Quantum Of Radiation

In 1905, Albert Einstein provided a revolutionary explanation for the photoelectric effect by proposing a new concept: that electromagnetic radiation (light) consists of discrete packets or quanta of energy, later called photons. The energy of each quantum is directly proportional to the frequency of the light, given by $E = h\nu$, where $h$ is Planck's constant (first introduced by Max Planck in the context of blackbody radiation).

Einstein proposed that in the photoelectric effect, an electron at the metal surface absorbs a single quantum of energy (photon) from the incident radiation. If the energy of this absorbed photon ($h\nu$) is greater than the minimum energy required to remove the electron from the metal surface (the work function, $\phi_0$), the electron is emitted. Any energy from the photon in excess of the work function is converted into the kinetic energy of the emitted electron. The most energetic electrons are those that absorb a photon at the surface and do not lose energy through collisions within the metal. Their maximum kinetic energy ($K_{max}$) is given by:

$$K_{max} = h\nu - \phi_0$$

This is known as Einstein's photoelectric equation. Electrons bound more tightly than the minimum work function (if any) or those that suffer collisions before escaping will have less than this maximum kinetic energy.

Einstein's equation elegantly explains all the experimental observations of the photoelectric effect:

• Dependence of $K_{max}$ on Frequency and Intensity: The equation $K_{max} = h\nu - \phi_0$ shows that $K_{max}$ depends linearly on the frequency ($\nu$) of the incident light and on the work function ($\phi_0$) of the metal. It is explicitly independent of the intensity of the radiation. This is because a single photon interacts with a single electron, and the photon's energy is solely determined by its frequency. Increasing intensity means increasing the *number* of photons, not the energy per photon.
• Existence of Threshold Frequency: For an electron to be emitted, its kinetic energy must be non-negative ($K_{max} \ge 0$). Therefore, $h\nu - \phi_0 \ge 0$, which implies $h\nu \ge \phi_0$ or $\nu \ge \phi_0/h$. This establishes a minimum frequency, the threshold frequency $\nu_0 = \phi_0/h$. If the incident frequency $\nu$ is below $\nu_0$, the photon energy $h\nu$ is less than $\phi_0$, and even if an electron absorbs this photon, it does not have enough energy to escape. Increasing intensity (more low-energy photons) does not help, as an electron absorbs only one photon at a time.
• Dependence of Photocurrent on Intensity: If the incident frequency $\nu$ is above the threshold frequency $\nu_0$, each photon has enough energy to potentially eject an electron. Increasing the intensity of light means increasing the number of photons incident per unit area per unit time. More incident photons lead to more interactions with electrons and thus more photoelectrons emitted per second, resulting in a higher photocurrent.
• Instantaneous Emission: The fundamental process in Einstein's picture is the absorption of a single photon by a single electron, which is considered an instantaneous event. As soon as a photon with sufficient energy strikes an electron at the surface, emission can occur. There is no need for the electron to accumulate energy over time. Low intensity simply means fewer photons arrive per second, resulting in a lower photocurrent, but the emission process for each successful photon absorption is still instantaneous.

Relating the stopping potential $V_0$ to $K_{max}$ ($K_{max} = eV_0$), Einstein's equation can be written as:

$$eV_0 = h\nu - \phi_0 \quad \text{for } \nu \ge \nu_0$$

or

$$V_0 = \left(\frac{h}{e}\right)\nu - \frac{\phi_0}{e}$$

This form directly matches the experimentally observed linear relationship between $V_0$ and $\nu$ (

Figure 11.5). The slope of the $V_0$ versus $\nu$ graph should be equal to $h/e$, and the intercept on the frequency axis gives the threshold frequency $\nu_0 = \phi_0/h$. Millikan's precise experiments measured this slope for various metals, and using the known value of the elementary charge $e$, he was able to determine Planck's constant $h$. The value he obtained was in excellent agreement with the value determined from unrelated phenomena like blackbody radiation. This provided strong experimental confirmation of Einstein's photoelectric equation and the photon concept.

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Particle Nature Of Light: The Photon

The successful explanation of the photoelectric effect by Einstein, based on the idea of discrete energy packets, provided strong evidence that light, when interacting with matter, behaves as if it is composed of particles. These energy quanta were later named photons.

Einstein further reasoned that if a light quantum possesses energy $E = h\nu$, it should also possess momentum. For a photon moving at the speed of light $c$, the momentum $p$ is related to energy by $E = pc$. Therefore, the momentum of a photon is $p = E/c = h\nu/c$. Since $\nu = c/\lambda$, the momentum can also be expressed as $p = h/\lambda$.

Possessing definite values of both energy and momentum is a characteristic property of particles. This solidified the idea that the light quantum could be associated with a particle-like entity. Further experimental evidence for the particle nature of light came from A.H. Compton's experiments in 1924 on the scattering of X-rays by electrons (Compton effect), where X-rays behaved as particles colliding with electrons, conserving both energy and momentum.

The key characteristics of photons, the particles of electromagnetic radiation, can be summarized as follows:

• In interactions with matter, electromagnetic radiation acts as though it's made up of particles called photons.
• Each photon of frequency $\nu$ (or wavelength $\lambda$) carries a specific amount of energy $E = h\nu = hc/\lambda$.
• Each photon also carries a specific momentum $p = h\nu/c = h/\lambda$.
• Photons travel at the speed of light, $c$, in vacuum.
• All photons of light with the same frequency (or wavelength) have the same energy and momentum, regardless of the intensity of the radiation. Higher intensity simply means a greater number of photons per second crossing a given area.
• Photons are electrically neutral and are not affected by electric or magnetic fields.
• In a collision between a photon and a material particle (like an electron), total energy and total momentum are conserved, consistent with collision principles. However, the number of photons involved in a process (like absorption or emission) may not be conserved.

Wave Nature Of Matter

Having established that radiation (like light) exhibits a dual nature – behaving sometimes as a wave and sometimes as a particle – a profound question arose: Does matter, typically considered to be made of particles (electrons, protons, etc.), also possess wave-like properties?

In 1924, Louis Victor de Broglie proposed a bold hypothesis based on the idea of symmetry in nature. He reasoned that if radiation, which primarily exhibits wave phenomena, can also show particle characteristics, then matter, which is primarily seen as particles, should similarly exhibit wave characteristics under appropriate conditions. De Broglie proposed that a wave is associated with every moving material particle, and the wavelength ($\lambda$) of this "matter wave" is related to the particle's momentum ($p$) by the same relationship that applies to photons:

$$\lambda = \frac{h}{p}$$

where $h$ is Planck's constant. For a particle of mass $m$ moving with velocity $v$, its momentum is $p = mv$. Thus, the de Broglie wavelength is given by:

$$\lambda = \frac{h}{mv}$$

This equation is known as the de Broglie relation and embodies the wave-particle duality of matter. The left side ($\lambda$) represents a wave attribute, while the right side (involving $m$ and $v$, or $p$) represents particle attributes. Planck's constant $h$ serves as the link between these two aspects.

De Broglie's hypothesis was revolutionary and needed experimental verification. The de Broglie wavelength is inversely proportional to the momentum ($mv$) of the particle. For macroscopic objects (large mass and/or velocity), the momentum is large, resulting in an extremely small de Broglie wavelength. For instance, a baseball or a bullet has a de Broglie wavelength far too small to be detected by any current experimental method. This is why we do not observe wave-like phenomena (like diffraction or interference) with everyday objects; they primarily exhibit particle behavior.

However, for microscopic particles like electrons, protons, neutrons, or atoms, the mass ($m$) is very small. Even with moderate velocities, their momentum ($p$) can be small enough to result in a de Broglie wavelength that is comparable to interatomic distances in crystals or spacings in diffraction gratings. In these cases, the wave nature of matter becomes significant and can be experimentally observed. The wave nature of electrons was first experimentally confirmed by Clinton Davisson and Lester Germer in 1927 through electron diffraction experiments, followed by George Paget Thomson (J.J. Thomson's son) shortly after. This experimental confirmation earned de Broglie the Nobel Prize in Physics in 1929.

The concept of matter waves is fundamental to quantum mechanics, where the behavior of particles is described by wave functions, and phenomena like electron diffraction through crystals are explained as constructive and destructive interference of these matter waves.

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