Dual Nature of Radiation and Matter

14.1 Introduction

• Reflection, refraction, interference, diffraction and polarization are explained by treating light as an electromagnetic wave: mutually perpendicular oscillating electric and magnetic fields, both perpendicular to the direction of propagation.
• Black body radiation could not be explained by the wave picture. Such phenomena appear when radiation interacts with matter and need quantum physics.
• Planck's quantization of energy: energy is emitted in packets, and at higher frequency the packet is larger. Atoms behave like tiny oscillators emitting radiation only in discrete packets $E = nh\nu$, emission occurring when the oscillator jumps from one quantized level to a lower one.
• Planck's model became the basis of Einstein's explanation of the photoelectric effect.

14.2 The Photoelectric Effect

• Photoelectric effect: the phenomenon of emission of electrons from a metal surface when radiation of appropriate frequency is incident on it.
• A surface that emits electrons when illuminated with appropriate radiation is a photosensitive surface. The emitted electrons are called photoelectrons (they are ordinary electrons; the name only records how they were released).
• Discovered by Heinrich Hertz in 1887 while producing electromagnetic waves by spark discharge.
• Zinc, cadmium and magnesium need ultraviolet radiation; alkali metals respond even to visible light.
• Two ways of getting electrical energy from light: photo-emissive effect (electrons ejected from a surface, as here) and photo-voltaic effect (a potential difference generated in a semiconductor, as in a solar cell).

Experimental set-up

• Evacuated glass tube with a quartz window, containing a photosensitive emitter plate E (cathode) and a collector plate C (anode).
• E and C are connected to a variable voltage source and an ammeter; a voltmeter reads the potential difference $V$ between them.
• Positive anode potential accelerates the photoelectrons (accelerating potential); negative anode potential retards them (retarding potential).
• Source S gives monochromatic light of sufficiently high frequency, wavelength $\leq 10^{-7}$ m.
• The quartz window matters: ordinary glass absorbs ultraviolet radiation.

The nine observations

• Photocurrent flows only if $\nu > \nu_0$, the threshold frequency, which is fixed for a given metal and different for different metals. Above $\nu_0$ electrons are emitted however feeble the light.
• Emission is instantaneous: the current starts within about $10^{-9}$ s and stops the moment the radiation is cut off. There is no time lag.
• For fixed $\nu$ and fixed accelerating potential $V$, the photocurrent increases linearly with intensity.
• For fixed incident radiation, the current first rises with the accelerating potential and then becomes constant at the saturation current $I_0$.
• Increasing the intensity increases $I_0$ proportionately: double the intensity, double the saturation current.
• $K.E._{max}$ of the photoelectrons depends on the emitter material and on the incident frequency, and is completely independent of the intensity.
• Increasingly negative collector potential reduces the current to zero at $-V_0$, the cut-off or stopping potential. $V_0$ is the same for any intensity at a given frequency, but different for different emitter materials.
• Changing $\nu$ at constant intensity leaves the saturation current unchanged but changes $V_0$; $V_0$ varies linearly with $\nu$.
• The graph of $V_0$ against $\nu$ is a straight line whose slope is the same for all metals but whose intercepts differ.
• Photocurrent, and hence the number of electrons, depends on intensity and not on frequency, so long as $\nu > \nu_0$ and the anode is positive.

Work function

• Work function $\phi_0$: the minimum amount of energy required to be provided to an electron to pull it out of the metal from the surface.
• It depends on the metal and on the nature of its surface, and is quoted in eV. What ionization energy is to an atom, the work function is to a solid.

• Highest work function in the table is gold, so gold needs the highest incident frequency to generate photocurrent. Potassium has the lowest, so it works with visible light and is the cathode material used in a street-light photocell.

Failure of the wave theory

• Numerical estimate in the textbook: for potassium of area $5\ \text{cm}^2$ under intensity $0.5\times10^{-4}\ \text{W m}^{-2}$, the wave picture gives a time lag of about $4.428\times10^{4}$ s, roughly half a day, against an observed lag of under $10^{-9}$ s.

Einstein's explanation

• Einstein (1905) extended Planck's quantization to all electromagnetic radiation. A quantum of light is a photon of energy $E = h\nu$.
• The photon gives all its energy to one electron and ceases to exist. Emission occurs only if $h\nu \geq \phi_0$, giving $\nu_0 = \phi_0/h$.
• Energy left over after escaping is the kinetic energy: $K.E._{max} = h\nu - \phi_0$, Einstein's photoelectric equation.
• Emitted electrons have a spread of energies because electrons in a solid do not all have the same energy, are ejected from different depths, and may suffer collisions on the way out. $K.E._{max}$ belongs to the most energetic one.
• With $K.E._{max} = eV_0$, the equation becomes $V_0 = (h/e)\nu - \phi_0/e$, which is why all the $V_0$ against $\nu$ lines have the same slope $h/e$ and different intercepts.
• Millikan's 1909 measurement of $e$, and of $h$ from this slope, matched Planck's value and settled the matter. Einstein received the Nobel Prize in 1921 and Millikan in 1923.

14.3 Wave-Particle Duality of Electromagnetic Radiation

• Compton (1923-24) sent monochromatic X-rays of wavelength $\lambda$ onto graphite and found the scattered intensity peaked at two wavelengths: the original $\lambda$ and a longer $\lambda'$.
• He treated it as a billiard-ball collision of an X-ray photon with an electron. Energy and momentum are transferred, so the scattered photon has lower energy, lower frequency and longer wavelength.
• Compton shift $\Delta\lambda = \dfrac{h}{m_e c}(1 - \cos\theta)$, depending only on the scattering angle $\theta$ and not on the incident wavelength. Letting $h \to 0$ recovers the wave-theory result.
• Photon facts: a photon carries energy and momentum; all photons of a given frequency have the same energy and momentum; photons are electrically neutral and are not deflected by electric or magnetic fields; in a photon-particle collision energy and momentum are conserved but the number of photons is not; a photon always moves with speed $c$ and is never at rest; its rest mass is zero in all frames.
• Wave nature shows in interference and diffraction, when slit width or slit separation is smaller than or comparable to the wavelength. Particle nature shows in black body radiation, the photoelectric effect and Compton scattering.
• Since neither picture alone covers all the observations, both must be kept: one dominates in some situations and the other in the rest. This is the wave-particle duality of electromagnetic radiation.

14.4 Photo Cell

• A photo cell is a device that uses the photoelectric effect to convert light energy into electrical energy.
• Construction: a semi-cylindrical photosensitive metal plate E as cathode and a wire loop collector C as anode, supported in an evacuated glass or quartz bulb, connected to an external circuit with a high tension battery B and a microammeter. The photosensitive material may instead be pasted as a thin film on the inner wall of the bulb.
• Working: light of suitable wavelength on the cathode ejects photoelectrons, the applied field drives them to the anode, and the microammeter reads the photocurrent.
• Uses: control systems, light-measuring devices, light meters in photographic cameras, switching street lights on and off, counting devices, burglar alarms and traffic-defaulter alarms (an interrupted ultraviolet beam stops the photocurrent and triggers the counter or alarm).

14.5 De Broglie Hypothesis

• De Broglie (1924) argued from the symmetry of nature: if radiation has a dual nature, matter should have one too.
• For a photon, $p = E/c = h\nu/c = h/\lambda$. De Broglie proposed that a moving material particle of total energy $E$ and momentum $p$ has a wave associated with it in the same way, with $\nu = E/h$ and $\lambda = h/p = h/mv$.
• These waves are matter waves and $\lambda$ is the de Broglie wavelength. Greater the momentum, shorter the wavelength.
• Useful forms: $\lambda = h/\sqrt{2mE_K}$, and $\lambda = h/\sqrt{2mqV}$ for a charge $q$ accelerated from rest through $V$ volt. For an electron this reduces to $\lambda = 1.228/\sqrt{V}$ nm.
• The relation holds for electrons, protons and charged ions, but not when $V$ is so large that the speed approaches $c$.
• A student of $45$ kg running at $8\ \text{km h}^{-1}$ has $\lambda = 6.63\times10^{-36}$ m and a $1200$ kg car at $60\ \text{km h}^{-1}$ has $\lambda = 3.32\times10^{-38}$ m. These are negligible against the size of the object and the width of the path, which is why we never notice the wave nature of macroscopic bodies.
• An electron of $100$ eV has $\lambda = 1.228$ angstrom, comparable to the size of an atom, so its wave nature is easy to see.

14.6 Davisson and Germer Experiment

• Purpose: to test experimentally whether electrons, which are particles, show diffraction, which is a property of waves.
• Set-up: an evacuated chamber containing an electron gun (tungsten filament F heated by battery B), a cylindrical anode that accelerates and collimates the beam, a nickel crystal target, a movable electron detector on a circular scale and a galvanometer.
• The scattered intensity was not uniform in all directions as classical theory predicted; it showed peaks (constructive interference) and troughs (destructive interference), that is, a diffraction pattern.
• Varying the accelerating potential from $44$ V to $68$ V, a sharp peak appeared at scattering angle $50^\circ$ for $V = 54$ V.
• Theory: $\lambda = 1.228/\sqrt{54} = 0.167$ nm. Experiment gave $0.165$ nm. The close agreement substantiated de Broglie's hypothesis and verified his relation.
• G. P. Thomson obtained the same conclusion in 1928 by diffracting electrons through metal foils. De Broglie got the Nobel Prize in 1929; Davisson, Germer and Thomson shared it in 1937. J. J. Thomson discovered the electron as a particle and his son G. P. Thomson showed it behaves as a wave.

14.7 Wave-Particle Duality of Matter

• Material particles show wave-like nature under certain circumstances; this is the wave-particle duality of matter.
• Waves are described classically by frequency and wave number, particles by mass and momentum. Duality means every moving particle has an associated frequency and wave number, and every wave has an associated energy and momentum.
• Quantum behaviour becomes noticeable only for microscopic objects, of size about $10^{-6}$ m or less. The small magnitude of $h$ fixes the scale at which quantum phenomena appear.
• Whenever the wavelength is small compared with the dimensions of slits or obstacles, or is too small to measure, Newtonian mechanics is enough.
• In conclusion: for both electromagnetic radiation and atomic or sub-atomic particles, the particle nature dominates during interaction with matter, while the wave nature dominates during travel through space, particularly when the confinement is of the same order as the associated wavelength.
• Three ways of supplying an electron the work function needed to escape: thermionic emission (heating to about $2000\ ^\circ$C), field emission (a strong field of about $10^{6}\ \text{V m}^{-1}$ at a metal tip) and photoelectric emission (shining ultraviolet or visible radiation).
• Electron microscope: resolution improves as the probing wavelength shortens. An electron beam of a few keV has a de Broglie wavelength far below an angstrom, so the resolution is several hundred times better than an optical microscope. Electrons also do not penetrate matter as light or X-rays do, are easily produced and controlled by electric and magnetic fields, and can be focused like light.

Distinctions the board asks for

Standard graphs to be able to draw

• Photocurrent against intensity, at fixed frequency and fixed accelerating potential: a straight line through the origin.
• Photocurrent against accelerating potential, at fixed frequency and three different intensities: three curves rising and flattening to three different saturation currents, but all cutting the voltage axis at the same stopping potential $-V_0$.
• Photocurrent against accelerating potential, at fixed intensity and three different frequencies: all curves flatten to the same saturation current, but the stopping potentials $-V_0^1$, $-V_0^2$, $-V_0^3$ are different, the largest belonging to the highest frequency.
• Stopping potential against frequency, for emitters of different metals: parallel straight lines, common slope $h/e$, different intercepts $\nu_0$ on the frequency axis and $-\phi_0/e$ on the $V_0$ axis.

Traps that cost marks

• Writing $K.E._{max} = h\nu$ and forgetting $\phi_0$. The equation is $K.E._{max} = h\nu - \phi_0$.
• Comparing an energy in eV with an energy in joule. Convert $\phi_0$ to joule (multiply by $1.6\times10^{-19}$) before subtracting it from $hc/\lambda$.
• Substituting $K.E._{max}$ in eV into $v_{max} = \sqrt{2\,K.E._{max}/m}$. It must be in joule, or the speed comes out wrong by a factor of about $10^{9}$ under the root.
• Saying photoemission needs high intensity. It needs high enough frequency; intensity only controls how many electrons come out.
• Saying stopping potential depends on intensity. It does not; it depends on the incident frequency and on the emitter material.
• With polychromatic light, the stopping potential is set by the shortest wavelength (highest frequency) present, not the average.
• Confusing the two threshold inequalities. Emission requires $\nu > \nu_0$ but $\lambda < \lambda_0$ — the wavelength condition reverses.
• Using $p = E/c$ for an electron. That expression is only for a massless particle moving at $c$, i.e. a photon. For an electron use $p = mv$ or $p = \sqrt{2mE_K}$.
• Forgetting that an electron at rest has $p = 0$, so its de Broglie wavelength is infinite, i.e. undefined. Only a moving particle has a de Broglie wavelength.
• Using $\lambda = 1.228/\sqrt{V}$ and then reporting the answer in angstrom. That formula gives nanometre; the angstrom form is $12.28/\sqrt{V}$.
• Applying $\lambda = h/\sqrt{2mqV}$ with $V$ in kilovolt without noting that the relation breaks down as the speed approaches $c$.
• Quoting the Davisson-Germer peak as being at $54^\circ$ and $50$ V. It is at scattering angle $50^\circ$ for an accelerating potential of $54$ V.
• Saying red light can never cause photoemission. It can, if the metal has a low enough work function; the correct answer is that any radiation of sufficient energy will do.
• Forgetting that the photocurrent stops the instant the light is switched off, since no photons remain to transfer energy.