Dual Nature of Radiation and Matter: Class 12 Physics · Chapter 11

11.1 Introduction

By now the case for light as a wave looks closed. Two slits give bright and dark fringes, a narrow gap spreads a beam, and a Polaroid can cut a reflection's glare, as you saw in Wave Optics. Maxwell's equations even predict the speed of the wave. Yet the light meter of a camera, the photomultiplier tube that counts single flashes in a particle detector, and the night-vision tube all work on an effect that the wave picture cannot explain. This chapter follows that puzzle to its answer, and then turns the answer round: if light can act like a stream of particles, can particles act like waves?

The puzzle grew out of experiments in the last quarter of the nineteenth century. In 1887 Heinrich Hertz made and detected electromagnetic waves, a triumph for Maxwell's wave theory. At the same time, physicists were passing electricity through gases at very low pressure, and these experiments kept producing surprises: X-rays (Wilhelm Röntgen, 1895) and, above all, the electron.

Seal two metal electrodes into a glass tube, pump out nearly all the air and connect a few thousand volts across them. Once the pressure is down to about a thousandth of a millimetre of mercury, roughly a millionth of an atmosphere, the glass at the far end from the negative electrode starts to glow. Its colour depends on the glass; ordinary soda glass glows green. Something is streaming out of the cathode and striking the glass. These cathode rays (कैथोड किरणें) were studied closely by William Crookes in the 1870s, and in 1879 he argued that they were streams of fast, negatively charged particles.

In 1897 J. J. Thomson tested the idea with a beam of cathode rays, an electric field between two plates and a magnetic field at right angles to both the beam and the electric field (Fig. 11.1). Each field alone bends the beam; set up together and adjusted, they can cancel. The cancellation gives the speed of the particles, and the bending by one field alone then gives their charge-to-mass ratio, or specific charge $e\mathrm{/}m$. The particles turned out to move at about one-tenth to one-fifth of the speed of light, $c=3\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$.

Two findings made this particle special. Its $e\mathrm{/}m$ came out the same whatever metal the cathode was made of and whatever gas was left in the tube. And negative particles knocked out of a metal by ultraviolet light, or boiled out of a white-hot filament, had exactly the same $e\mathrm{/}m$. One kind of particle, produced in very different ways from very different materials, had to be a common ingredient of all matter. Thomson called it the electron; the presently accepted value of its specific charge is $e\mathrm{/}m=1.76\times {10}^{11}\mathrm{C}\mathrm{/}\mathrm{k}\mathrm{g}$. He received the Nobel Prize in Physics in 1906.

The charge itself came from R. A. Millikan's oil-drop experiment, published in 1913. Millikan timed tiny charged oil drops as they rose and fell between two charged plates, and found that the charge on every drop was a whole-number multiple of one smallest charge, $e=1.602\times {10}^{-19}\mathrm{C}$. Charge is quantised, as you assumed in Chapter 1 of Part I. Dividing $e$ by $e\mathrm{/}m$ then gave the electron's mass, about $9.11\times {10}^{-31}\mathrm{k}\mathrm{g}$, nearly two thousand times lighter than the lightest atom.

So, by the early 1900s, the electron was a particle with a definite charge and mass, and light was a wave. The rest of this chapter shows that each of these statements is only half the story: radiation and matter both have a dual nature (द्वैत प्रकृति).

11.2 Electron emission

A copper wire is crowded with free electrons; that is why it conducts so well. Yet a wire lying on a table does not spray electrons into the room. Deep inside the metal, each free electron has positive ions on every side and their pulls cancel. At the surface that balance breaks. An electron that strays outside leaves the metal slightly positive, and the attraction of the ions it left behind drags it back.

To leave for good, an electron must be given a certain minimum amount of energy. This least energy needed to free an electron from a metal surface is the work function (कार्य फलन) of the metal, written ${\varphi }_0$. Its value depends on the metal and also on the condition of the surface: a thin oxide layer, a film of adsorbed gas or a trace of another metal can change it noticeably.

Work functions are a few times ${10}^{-19}\mathrm{J}$, clumsy numbers to carry around. Atomic and nuclear physics therefore uses a smaller unit, the electron volt: the kinetic energy an electron gains when it is accelerated from rest through a potential difference of $1\mathrm{V}$. Since the energy gained is $qV$,

On this scale work functions lie between about $2\mathrm{e}\mathrm{V}$ for caesium and nearly $6\mathrm{e}\mathrm{V}$ for platinum. An electron accelerated through $1000\mathrm{V}$ gains $1000\mathrm{e}\mathrm{V}$, or $1\mathrm{k}\mathrm{e}\mathrm{V}$, whatever the size of the apparatus.

How can a metal's electrons be given that energy? There are three practical ways, compared in Fig. 11.2.

1. Heating. At high temperature some free electrons gain enough thermal energy to cross the surface. This thermionic emission supplies the electrons in X-ray tubes and electron guns, from a glowing tungsten filament.
2. A very strong electric field. A field of order ${10}^8\mathrm{V}\mathrm{/}\mathrm{m}$ at a metal surface can pull electrons straight out. This is field emission; the very sharp tips used as electron sources in some electron microscopes work this way.
3. Light. Light of a suitable frequency falling on a metal can free electrons from it. This is photoelectric emission, and the electrons it frees are called photoelectrons (प्रकाशिक इलेक्ट्रॉन).

11.3 Photoelectric effect

Of the three ways of freeing electrons, the third turned out to hold the deepest lesson. It was first noticed by accident.

11.3.1 Hertz’s observations

In 1887 Heinrich Hertz (1857–1894) was sending electromagnetic waves across his laboratory. His detector was a loop of wire with a tiny gap, and a received wave showed itself as a spark across the gap. Hertz noticed that the detector sparked more readily when ultraviolet light, from his transmitter's spark or from an arc lamp, fell on the metal near the gap. Shielding the gap from the light made the sparks weaker. We now know what was happening. The ultraviolet light handed certain surface electrons the energy they needed to break away from the metal's ions, and these free charges helped each spark to start.

11.3.2 Hallwachs’ and Lenard’s observations

Wilhelm Hallwachs followed the clue in 1888 with a clean zinc plate fixed to the top of a gold-leaf electroscope (Fig. 11.3). He found three things. A negatively charged plate lost its charge when ultraviolet light fell on it. An uncharged plate became positively charged under the same light. And a plate that was already positive became more positive. Each result says the same thing: ultraviolet light drives negative charge out of zinc.

Philipp Lenard (1862–1947) took the measurements inside a vacuum. He sealed two metal plates in an evacuated glass tube and let ultraviolet light fall on one of them, the emitter, while the other, the collector, was kept positive. A current flowed round the circuit for exactly as long as the light was on. Electrons released at the emitter were drawn across the vacuum to the positive collector, so light falling on a metal could drive a current through an external circuit. Their later measurements asked three questions of this photocurrent: how does it respond to the collector's potential, to the light's frequency, and to its brightness?

One result stood out. For light below a certain frequency no electrons came out at all, however long the plate was lit. This minimum frequency, the threshold frequency (देहली आवृत्ति), is a property of the emitter, and it explains why metals behave so differently. For zinc, cadmium or magnesium it lies in the ultraviolet, so visible light does nothing. For the alkali metals (lithium, sodium, potassium, rubidium, caesium) it lies in the visible, so even ordinary lamplight frees electrons.

11.4 Experimental study of photoelectric effect

To turn these observations into laws, we need a tube in which every variable can be set and measured. Fig. 11.4 shows one design. The glass bulb is evacuated, so that the electrons do not collide with gas molecules on their way. Inside is a photosensitive emitter C, here a curved sheet, and a collector A, here a rod. The light enters through a quartz window, because ordinary glass absorbs ultraviolet light.

Each quantity we want to vary has its own control. Colour, and so frequency: swap the filter in front of the lamp, or the lamp itself. Brightness: slide the lamp along a bench, since intensity falls off with distance. Potential: a dc supply with a reversing switch (often called a commutator) holds the collector at a chosen potential $V$ relative to the emitter, either positive or negative. Two meters complete the circuit: a voltmeter for $V$, and a microammeter for the photocurrent, which is rarely more than a few microamperes.

Four things can now be varied one at a time: the brightness of the light, its frequency, the collector's potential and the emitter's material. The next three subsections take them in turn, holding everything else fixed.

11.4.1 Effect of intensity of light on photocurrent

Keep the collector positive, keep the colour of the light fixed and change only the intensity. A simple way is to move the lamp: intensity falls off as the inverse square of distance, so halving the distance makes the light four times as intense. The microammeter then reads four times as much (Fig. 11.5). Repeat for several distances and the readings fall on a straight line through the origin: the photocurrent is directly proportional to the intensity.

Every electron that reaches the collector carries the same charge $e$, so the current counts electrons: $n$ electrons per second give $I=ne$. A current proportional to intensity therefore tells us that the rate at which photoelectrons are released rises in proportion to the intensity of the light.

11.4.2 Effect of potential on photoelectric current

Now fix both the frequency and the intensity of the light and vary the collector's potential $V$. Start with the collector positive and raise $V$ step by step. The photocurrent grows, because a stronger attraction gathers electrons that would otherwise have strayed, and then it levels off. Beyond a certain potential, raising $V$ further changes nothing: every electron emitted is already being collected. This largest current is the saturation current (संतृप्त धारा).

Even at $V=0$ some current flows, because the electrons leave the emitter with kinetic energy of their own and many reach the collector unaided. To find out how much energy they have, reverse the supply so that the collector is negative with respect to the emitter. The field now pushes electrons back towards the emitter, and only those with enough kinetic energy can climb the potential difference to reach the collector. As the collector is made more negative, the current falls quickly, and at one sharply defined value of the retarding potential it becomes zero.

Here is how to read it. An electron that travels from emitter to collector against a retarding potential difference $V$ loses kinetic energy $eV$. The photoelectrons do not all leave with the same energy, so a small retarding potential stops only the slowest. The current vanishes when the retarding potential is just enough to stop the fastest electrons, those with the maximum kinetic energy $K_{\max }$. Therefore

A stopping potential of $0.8\mathrm{V}$ means that the fastest photoelectrons leave with $0.8\mathrm{e}\mathrm{V}$: the electron volt makes this conversion free. Now repeat the whole run with brighter light of the same frequency (Fig. 11.6). The saturation current rises in proportion to the intensity, since more electrons are emitted each second. But the curve reaches zero at exactly the same stopping potential. For a given frequency, the stopping potential does not depend on the intensity: brighter light frees more electrons, not faster ones. The maximum kinetic energy depends on the light's frequency and on the emitter, not on how bright the light is.

11.4.3 Effect of frequency of incident radiation on stopping potential

Finally, change the frequency of the light. To compare like with like, adjust the intensity at each frequency until the saturation current is the same. The current–potential curves then differ in only one place: where they meet the axis (Fig. 11.7). The higher the frequency, the larger the stopping potential, so the more negative the collector must be made to stop the current. Higher-frequency light gives faster photoelectrons.

Now plot the stopping potential against the frequency for one emitter (Fig. 11.8). The points fall on a straight line, and the line meets the frequency axis at a definite frequency ${\nu }_0$ where the stopping potential is zero. Below ${\nu }_0$ there is nothing to stop, because no electrons are emitted. Repeat for another metal and you get another straight line, parallel to the first but crossing the axis at a different frequency.

Not every surface is equally willing. Copper gives photoelectrons under ultraviolet light but none at all under green or red light, however bright, while selenium responds more readily than either copper or zinc.

One more fact completes the picture. Whenever the frequency is above threshold, emission begins the moment the light is switched on, with no delay that any experiment has detected, even when the light is extremely dim. Measurements put any delay below about ${10}^{-9}\mathrm{s}$.

Together these experiments give four laws of photoelectric emission, for a given emitter:

1. Above the threshold frequency, the photocurrent is directly proportional to the intensity of the light (Fig. 11.5).
2. The saturation current is proportional to the intensity, but the stopping potential, and so $K_{\max }$, does not depend on the intensity (Fig. 11.6).
3. There is a threshold frequency below which no photoelectrons are emitted, however intense the light. Above it, the stopping potential, and so $K_{\max }$, increases linearly with frequency (Fig. 11.8).
4. Emission is instantaneous: it starts within about ${10}^{-9}\mathrm{s}$ of the light arriving, even in very dim light.

11.5 Photoelectric effect and wave theory of light

By 1900 the wave theory of light seemed beyond doubt. It explained interference, diffraction and polarisation, and Maxwell's theory identified light as an electromagnetic wave. In that picture the energy of a light beam is spread smoothly over each wavefront and flows continuously, like water from a hose rather than like a stream of pebbles. Let us test this picture against the four laws.

A brighter wave has stronger electric and magnetic fields, so it should shake each surface electron harder and give it more energy. The maximum kinetic energy of the photoelectrons ought to grow with intensity. Since the energy simply accumulates, any frequency should work if the light is bright enough or shines for long enough, so there should be no threshold. Both predictions contradict the experiments (laws 2 and 3).

The timing is just as damaging. A wave delivers its energy over the whole illuminated area, shared among a vast number of electrons, so each electron's share arrives slowly. The example below estimates the wait. It comes out at hours for dim light, against the observed ${10}^{-9}\mathrm{s}$. In short, the wave picture cannot account for even the simplest facts of photoemission (Fig. 11.10).

11.6 Einstein’s photoelectric equation: energy quantum of radiation

In 1905 Albert Einstein proposed a radical way out. In 1900 Max Planck had found that he could explain the radiation from hot bodies only by assuming that energy passed between a hot body and its radiation in steps of $h\nu$, where $\nu$ is the frequency and $h$ is a new constant of nature, now called Planck's constant. Einstein went further. He suggested that the energy of radiation itself is made of separate lumps, or quanta, each of energy $h\nu$. Light is not absorbed as a smooth flow: an electron in the metal takes in one whole quantum, or nothing.

Fig. 11.11 shows the energy budget for potassium, whose work function is $2.30\mathrm{e}\mathrm{V}$. In this picture, for light of one fixed frequency, intensity just counts quanta: how many arrive on each square metre every second. Making the light brighter sends more quanta; it does not make any one quantum bigger.

Now check the equation against each of the four laws.

1. Frequency, not intensity, sets $K_{\max }$. Each emission is one quantum absorbed by one electron, so by Eq. (11.2) $K_{\max }$ depends on $\nu$ and on the metal, rising linearly with $\nu$. How many quanta arrive per second, which is what intensity measures, does not enter.
2. There is a threshold. Kinetic energy cannot be negative, so emission needs $h\nu \ge {\varphi }_0$, that is $\nu \ge {\nu }_0$ with ${\nu }_0$ given by Eq. (11.3) below. A metal with a larger work function needs a higher threshold frequency. Below ${\nu }_0$ no single quantum is big enough, and the quanta cannot pool their energy, so turning up the brightness, or waiting, achieves nothing.
3. Current is proportional to intensity. More quanta per second means more electrons absorbing one each, so more photoelectrons per second (for $\nu >{\nu }_0$).
4. No delay. Absorbing a quantum is a single, instantaneous event. Dim light means fewer such events per second, not slower ones, so the first electrons appear at once.

Combining Eq. (11.2) with $K_{\max }=e{V}_0$ from Eq. (11.1) gives a prediction that can be tested with a voltmeter, valid for $\nu \ge {\nu }_0$:

This is a strong prediction. A graph of $V_0$ against $\nu$ must be a straight line, its slope must be $h\mathrm{/}e$ for every metal, it must cross the frequency axis at ${\nu }_0$, and extended back to $\nu =0$ it must cut the potential axis at $-{\varphi }_0\mathrm{/}e$. Look again at the last step of Fig. 11.8: that is exactly what is observed.

The sternest test came from someone who expected the equation to fail. Robert Millikan spent ten years, 1906 to 1916, on it, building an apparatus that shaved fresh, clean surfaces of alkali metals inside a vacuum. To his surprise the graphs came out straight and parallel, exactly as Eq. (11.4) demands. Their slope multiplied by $e$ gave $h$; for sodium it matched the constant Planck had extracted from hot-body radiation, an entirely different phenomenon. Today's value is

Two independent routes to the same $h$, and work functions that matched those from other methods, left little room for doubt. Einstein's quanta were accepted, though many physicists stayed uneasy about what they meant for the wave theory.

11.7 Particle nature of light: the photon

Energy delivered in lumps of $h\nu$ does not by itself make light a stream of particles. What would? A cricket ball carries momentum as well as energy and hands both over when it hits something. Maxwell's theory already says that an electromagnetic wave carrying energy $E$ also carries momentum $E\mathrm{/}c$ (Part I, Chapter 8). Einstein applied this to a single quantum: each one must carry momentum $h\nu \mathrm{/}c$. In terms of wavelength,

With a definite energy and a definite momentum, each quantum behaves like a particle; the name photon for it was coined in 1926. The clinching evidence came in the early 1920s from Arthur Compton, who scattered X-rays off electrons. The scattered X-rays had longer wavelengths, by exactly the amount expected if each photon had collided with an electron like one billiard ball with another, sharing energy and momentum. Einstein received the 1921 Nobel Prize, Millikan the 1923 prize (for the electron's charge and for his photoelectric work), and Compton the 1927 prize.

Here is what the photon picture says, property by property.

• Energy and momentum. A photon of frequency $\nu$ carries $E=h\nu$ and $p=h\nu \mathrm{/}c=h\mathrm{/}\lambda$, and in vacuum it always moves at $c$.
• Identical photons. Light of a single frequency consists of identical photons. Brightness tells you how many cross each square metre per second, not how big each one is.
• No charge. Photons are electrically neutral, so neither electric nor magnetic fields can bend their paths.
• Collisions. When a photon meets an electron or another particle, energy and momentum are both conserved, but the photon count is not: photons can be absorbed, and new ones created.
• Where it applies. These particle properties show up when light is emitted, absorbed or scattered by matter; while it travels, interference still needs the wave description.

11.8 Wave nature of matter

Look back over this chapter and Wave Optics, and light seems to change character with the experiment. Fringes and diffraction patterns can only be built from waves; the photoelectric effect and Compton scattering, in which light hands over energy and momentum, can only be explained with photons. Often one device needs both descriptions at once. In a phone camera the lens forms the image, and how sharp that image can be is limited by diffraction, a wave effect. Each pixel of the sensor, though, works on the photoelectric idea: arriving photons free electrons inside the silicon, at most one electron per photon, and the camera counts them (Fig. 11.13).

Louis de Broglie turned the argument round in 1924. Light, long called a wave, had turned out to carry momentum in particle-sized parcels; perhaps electrons, long called particles, carry a wave. His reason was a belief in symmetry. Matter and radiation are the two great ingredients of the universe, and it seemed unlikely that only one of them should lead a double life. He proposed that any particle with momentum $p$ is accompanied by a wave of wavelength

Equation (11.5) is the de Broglie relation, and $\lambda$ is the particle's de Broglie wavelength (दे ब्रॉग्ली तरंगदैर्ध्य), the wavelength of its matter wave. Read the two sides separately and the duality is in plain view: $\lambda$ is something only a wave can have, $p$ something we picture as belonging to a moving lump, and the tiny constant $h$ converts one into the other.

For electrons and atoms, the relation was in 1924 a bold guess waiting for experiment. For light it can be checked at once. A photon of frequency $\nu$ has momentum

so de Broglie's recipe gives it the wavelength

nothing other than the familiar wavelength of the light itself. The new rule contains the old one.

Why do we never notice the waves of moving objects? Because $h$ is so small, the wavelength is minute unless the mass and speed are tiny too. A raindrop of mass $4\mathrm{m}\mathrm{g}$ falling at $9\mathrm{m}\mathrm{/}\mathrm{s}$ has $\lambda =h\mathrm{/}mv\approx 1.8\times {10}^{-29}\mathrm{m}$, some ${10}^{14}$ times smaller than a proton. No slit or grating could ever reveal it. An electron is a different story: its mass is so small that its de Broglie wavelength is comparable with the spacing of atoms in a crystal, and then the wave behaviour is real and measurable (Fig. 11.14). Note too that $\lambda$ depends only on momentum, not on the charge or the kind of particle.