Dual Nature of Radiation and Matter

11.1 Introduction: how the electron was found

• Discharge of electricity through gases at low pressure (about $0.001$ mm of mercury) produces cathode rays. Crookes (1870) discovered them; he suggested in 1879 that they are streams of fast negatively charged particles.
• J. J. Thomson (1897) applied mutually perpendicular electric and magnetic fields across a discharge tube and measured the speed and the specific charge $e/m$ of these particles. Speeds came out as $0.1c$ to $0.2c$; the accepted value of $e/m$ is $1.76\times10^{11}$ C kg$^{-1}$.
• $e/m$ was found to be independent of the cathode material and of the gas in the tube — proof that these particles are universal constituents of matter. Thomson named them electrons and received the 1906 Nobel Prize.
• Millikan's oil-drop experiment (1913) showed that the charge on a drop is always an integral multiple of $1.602\times10^{-19}$ C, so electric charge is quantised. Knowing $e$ and $e/m$ gives the electron mass $m = 9.1\times10^{-31}$ kg.
• Roentgen discovered X-rays in 1895. Particles emitted by UV-irradiated metals and by hot metals were found to have the same $e/m$ as cathode rays — the same particle in every case.

Electron emission and work function

• Metals have free electrons, but they cannot leave the surface: an escaping electron leaves the surface positively charged, and is pulled back.
• Work function $\phi_0$: the minimum energy required by an electron to escape from the metal surface. Measured in eV; $1$ eV $=1.6\times10^{-19}$ J. It depends on the metal and on the nature of its surface.
• Three ways of supplying that energy:

Photoelectric effect: the discovery

• Photoelectric effect: the emission of electrons from a metal surface when light of suitable frequency falls on it. The emitted electrons are called photoelectrons.
• Hertz (1887): high-voltage sparks across his detector loop were enhanced when the emitter plate was illuminated by ultraviolet light from an arc lamp.
• Hallwachs (1888): a negatively charged zinc plate connected to an electroscope lost its charge under UV light; an uncharged plate became positively charged; a positively charged plate became more positive. Conclusion: negatively charged particles are emitted.
• Lenard (1886–1902): UV light on the emitter plate of an evacuated tube with two electrodes drove a current; the current stopped the instant the light was stopped.
• Below a certain minimum frequency — the threshold frequency $\nu_0$, characteristic of the emitter material — nothing is emitted, however intense the light.
• Zinc, cadmium and magnesium respond only to ultraviolet light. The alkali metals lithium, sodium, potassium, caesium and rubidium are sensitive even to visible light.

The experimental arrangement

• An evacuated glass or quartz tube contains a photosensitive plate C (the emitter/cathode) and a metal plate A (the collector/anode). The quartz window W lets ultraviolet light through.
• Monochromatic light from source S passes through W onto C. Electrons emitted from C are collected by A.
• A battery with a sliding contact sets the potential difference between C and A; a commutator reverses the polarity, so A can be made positive or negative with respect to C.
• A voltmeter reads the plate potential; a microammeter reads the photocurrent.
• The frequency is changed with coloured filters; the intensity is changed by moving the source nearer or further away.

The three graphs CBSE asks for

• Photocurrent versus intensity (collector positive, frequency and potential fixed): a straight line through the origin. Photocurrent is directly proportional to intensity, because the number of photoelectrons per second is proportional to the number of photons per second.
• Photocurrent versus collector plate potential: as the positive potential of A is raised, the current rises and then flattens at the saturation current, when every emitted electron reaches A. As A is made negative, the current falls and drops sharply to zero at the stopping potential $-V_0$. For a fixed frequency and increasing intensities $I_1 < I_2 < I_3$: three curves of the same shape, all cutting the potential axis at the same $-V_0$, saturating at successively higher currents. For a fixed intensity and increasing frequencies $\nu_1 < \nu_2 < \nu_3$: three curves saturating at the same current, but cutting the potential axis at successively more negative stopping potentials.
• Stopping potential versus frequency: a straight line, $V_0 = (h/e)\nu - \phi_0/e$. Slope $h/e$, the same for every material; it meets the $\nu$-axis at $\nu_0$ and the $V_0$-axis at $-\phi_0/e$. Two different metals give two parallel lines; the line further to the right has the greater $\nu_0$, hence the greater work function.
• Related straight lines: $K_{max}$ versus $\nu$ has slope $h$ and $\nu$-intercept $\nu_0$; $V_0$ versus $1/\lambda$ has slope $hc/e$; $v_m^2$ versus $1/\lambda$ has slope $2hc/m$.

The four experimental observations (learn this list)

• For a given material and a frequency above threshold, the photocurrent is directly proportional to the intensity of the incident light.
• For a given material and frequency, the saturation current is proportional to intensity, while the stopping potential is independent of intensity.
• For a given material there is a threshold frequency $\nu_0$ below which no emission occurs, however intense the light. Above $\nu_0$, the stopping potential — equivalently $K_{max}$ — increases linearly with frequency and is independent of intensity.
• Photoelectric emission is instantaneous: the time lag is about $10^{-9}$ s or less, even for very dim light.

Definitions worth writing verbatim

• Threshold frequency: the minimum frequency of incident radiation below which photoelectric emission does not take place, however large the intensity of light may be.
• Stopping potential: the minimum negative (retarding) potential given to the collector plate with respect to the emitter at which the photoelectric current becomes zero.
• Saturation current: the maximum photocurrent, reached when all the photoelectrons emitted per second are collected.
• Intensity, in the photon picture: the light energy — the number of photons multiplied by the energy of each photon — incident per unit area per unit time. SI unit W m$^{-2}$.
• Work function: the minimum energy required by an electron to escape from the metal surface.

Why the wave theory fails

• The stopping potential — independent of intensity, dependent on frequency — is the crucial discriminator between the wave picture and the photon picture.

Einstein's photoelectric equation (1905)

• Radiation energy comes in discrete quanta, each of energy $h\nu$. In photoemission a single electron absorbs a single quantum, all of it, at once.

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

• It explains all four observations at once: $K_{max}$ is linear in $\nu$ and free of intensity; emission needs $h\nu > \phi_0$, i.e. $\nu > \nu_0 = \phi_0/h$; intensity fixes the number of quanta and hence the current; and one-photon-one-electron absorption is instantaneous.
• Electrons deeper inside the metal are more tightly bound and emerge with less than $K_{max}$, so photoelectrons come out with a spread of energies from $0$ to $K_{max}$.
• Millikan set out (1906–1916) to disprove the equation, measured the slope of the $V_0$–$\nu$ line for sodium, obtained $h = 6.626\times10^{-34}$ J s in agreement with Planck's value, and ended up proving it.

The photon picture: five properties

• In its interaction with matter, radiation behaves as if made of particles called photons.
• Each photon has energy $E = h\nu$, momentum $p = h\nu/c = h/\lambda$, and speed $c$.
• All photons of a given frequency carry the same energy and momentum whatever the intensity. Raising the intensity increases only the number of photons crossing unit area per second.
• Photons are electrically neutral and are not deflected by electric or magnetic fields.
• In a photon-particle collision, total energy and total momentum are conserved, but the number of photons need not be — a photon may be absorbed or created.
• Compton's 1924 experiment on the scattering of X-rays by electrons confirmed the particle behaviour of light. Einstein received the 1921 Nobel Prize (partly for the photoelectric effect) and Millikan the 1923 Nobel Prize.

Wave nature of matter: de Broglie

• de Broglie's argument (1924): nature is symmetrical, and matter and radiation are the two basic entities; since radiation shows a dual character, matter should too. So a wave should be associated with every moving material particle.

$$\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$$

• The left side is a wave attribute, the right side a particle attribute, and Planck's constant links them — the dual nature of matter is contained in this one relation.
• The de Broglie wavelength is independent of the charge and the nature of the particle; it depends only on the momentum.
• Consistency with radiation: for a photon $p = h\nu/c$, so $\lambda = h/p = c/\nu$, exactly the wavelength of the radiation.
• Why we do not see matter waves: a $0.12$ kg ball at $20$ m s$^{-1}$ has $p = 2.40$ kg m s$^{-1}$ and $\lambda = 2.76\times10^{-34}$ m — far beyond measurement. An electron at $5.4\times10^{6}$ m s$^{-1}$ has $\lambda = 0.135$ nm, comparable to X-ray wavelengths and to the spacing of atomic planes in a crystal, so it is measurable.
• The phase velocity of a matter wave has no physical significance; the group velocity does, and equals the velocity of the particle.

Standard ratio results

Numbers CBSE reuses

• Board papers have also quoted Na as $1.92$ eV and K as $2.15$ eV in one 2018 question; always use the value printed in the paper, not a remembered one.

Traps that cost marks

• Stopping potential depends on frequency and on the material only. Doubling the intensity, or moving the lamp closer, changes nothing about $V_0$ or $K_{max}$ — only the current changes. This single point is examined nearly every year.
• Moving a source closer or further changes intensity, not frequency. A metal that would not emit still will not emit.
• Threshold is a condition on frequency, so on wavelength it reverses: emission needs $\nu \ge \nu_0$ but $\lambda \le \lambda_0$. A longer wavelength is a lower frequency.
• "Greater threshold frequency" means "greater work function" and "smaller threshold wavelength". Keep the three straight.
• The slope of the $V_0$–$\nu$ line is $h/e$ and is the same for all metals. Only the intercept changes with the material. Do not say the slope depends on the metal.
• Do not write "the kinetic energy of photoelectrons is $h\nu-\phi_0$". It is the maximum kinetic energy; electrons emerge with a spread of energies from zero up to that value, because they are not all equally bound.
• Photocurrent is proportional to intensity, not to frequency; $K_{max}$ is proportional to frequency (linearly, with an offset), not to intensity. Swapping these is the classic lost mark.
• $K_{max} = h\nu - \phi_0$ is linear in $\nu$ but not proportional to it: doubling $\nu$ gives $2K+\phi_0$, which is more than $2K$.
• $eV_0$ in joule equals $K_{max}$ in joule; if $V_0$ is in volt then $K_{max}$ in eV is the same number. Do not multiply by $e$ twice.
• Work function must be converted to joule before dividing by $h$, or else use $h = 4.14\times10^{-15}$ eV s throughout. Mixing units here wrecks the answer.
• de Broglie wavelength applies to every moving particle, not only to charged ones, and does not depend on charge. But $\lambda = h/\sqrt{2mqV}$ does, because that route uses $K = qV$.
• Do not use $\lambda = h/\sqrt{2mqV}$ for a neutral particle or for a particle whose kinetic energy was not acquired from an accelerating potential.
• $\lambda = 12.27/\sqrt V$ Å is for an electron only. Do not apply it to protons or alpha particles.
• A photon has zero rest mass but non-zero momentum $h/\lambda$; never write $p = mv$ for a photon.
• Photons are neutral, so they are not deflected by electric or magnetic fields — a favourite one-mark statement.
• On a $p$ versus $\lambda$ graph the curve is a rectangular hyperbola, not a straight line, since $p = h/\lambda$.
• In the photocurrent versus potential graph, remember the current is not zero at zero potential: some electrons reach the collector on their own kinetic energy. The curve meets the axis at the negative stopping potential.
• State intensity in the photon picture as energy per unit area per unit time, and give the unit W m$^{-2}$ when asked for it.