Dual Nature of Radiation and Matter: NEET notes

1. Electron emission

NCERT §11.1, §11.2

• Late in the nineteenth century, experiments on electric discharge through gases at low pressure led to several key discoveries. At a pressure of about 0.001 mm of mercury a discharge passes between the electrodes, and the glass facing the cathode glows; for soda glass the glow is yellowish-green.
• William Crookes found these cathode rays in 1870 and in 1879 proposed that they were streams of fast, negatively charged particles. J. J. Thomson confirmed this: using crossed electric and magnetic fields he measured their speed and their specific charge e/m.
• These particles travelled at roughly a tenth to a fifth of light's speed. Their measured e/m, 1.76 × 10¹¹ C/kg, came out the same whatever metal made the cathode and whatever gas filled the tube, so the particles are universal. Roentgen discovered X-rays in 1895, and Thomson named the particles electrons in 1897.
• Particles given off by metals under ultraviolet light, and by very hot metals, have the same e/m as cathode-ray particles: all of them are electrons. In 1913 Millikan's oil-drop experiment showed that charge always comes in whole multiples of 1.602 × 10⁻¹⁹ C, so charge is quantised; e and e/m together give the electron's mass.
• Free electrons inside a metal cannot simply leave it. An electron that starts to leave makes the surface positive, and the pull of the ions draws it back.
• The least energy an electron must be given to escape from the surface is the work function φ₀ of the metal. It depends on the metal and on the state of its surface, and it is usually quoted in electron volts: 1 eV, the energy an electron gains across a potential difference of 1 V, equals 1.602 × 10⁻¹⁹ J.
• The energy can be supplied in three ways. Thermionic emission: heating gives the free electrons enough thermal energy to escape. Field emission: a very strong electric field, of the order of 10⁸ V m⁻¹, pulls electrons out, as in a spark plug. Photoelectric emission: light of suitable frequency ejects electrons, called photoelectrons.

2. Discovery of the photoelectric effect

NCERT §11.3

• Heinrich Hertz came across photoelectric emission in 1887 while producing electromagnetic waves with a spark discharge. The sparks across his detector loop grew stronger when ultraviolet light from an arc lamp fell on the emitter plate.
• The light helps free charged particles, which we now know to be electrons, to leave the surface: electrons near the surface take in enough energy from the light to overcome the attraction of the positive ions.
• Wilhelm Hallwachs and Philipp Lenard studied the effect in detail between 1886 and 1902. Lenard shone ultraviolet light on one of two plates sealed in an evacuated tube: a current flowed in the circuit, and it stopped the moment the light was cut off.
• Hallwachs, in 1888, joined a zinc plate to an electroscope. A negatively charged plate lost its charge under ultraviolet light; an uncharged plate became positive; a positively charged plate became more positive. So the light was driving negative particles out of the zinc.
• Below a certain minimum frequency of light, the threshold frequency, no electrons came out at all. The threshold depends on the material of the emitter.
• Zinc, cadmium and magnesium need short-wavelength ultraviolet light before they give off electrons. The alkali metals (lithium, sodium, potassium, caesium, rubidium) give them off even in visible light.
• Once the electron was known, the emitted particles were called photoelectrons and the whole phenomenon the photoelectric effect.

3. The photoelectric experiment

NCERT §11.4, §11.4.1

• The apparatus is an evacuated glass or quartz tube holding a thin photosensitive plate C, the emitter, and a metal plate A, the collector. Monochromatic light from a source S enters through a quartz window W, which lets ultraviolet through, and falls on C.
• A battery keeps a potential difference between C and A that can be varied, and a commutator reverses it, so A can be made positive or negative with respect to C. A voltmeter reads the potential difference and a microammeter the photocurrent.
• When A is positive, it attracts the emitted electrons, and a current flows round the circuit. Four things can be varied: the intensity of the light, its frequency, the potential difference between A and C, and the material of C.
• Coloured filters or coloured glass in the light's path change its frequency. Moving the source nearer to or farther from the emitter changes its intensity.
• Effect of intensity: with the frequency and the accelerating potential held fixed, the photocurrent rises in direct proportion to the intensity of the light (a straight line through the origin, Fig. 11.2).
• Each electron that crosses the tube adds to the photocurrent, so the current counts photoelectrons per second. Doubling the intensity doubles that count: emission rate is directly proportional to intensity.

4. Saturation current and stopping potential

NCERT §11.4.2

• Keep the frequency ν and intensity fixed and make A more and more positive. The photocurrent grows, then levels off: at a large enough accelerating potential every electron emitted by C reaches A. This largest current is the saturation current, and raising the potential further does not increase it.
• Now make A negative with respect to C. The electrons are pushed back, and only the more energetic ones reach A, so the current falls quickly.
• At one sharply defined negative potential of A the current becomes zero. This smallest retarding potential that stops the photocurrent, for a given frequency, is the cut-off or stopping potential V₀.
• The photoelectrons do not all leave with the same energy. The stopping potential is just enough to turn back the fastest of them, so the maximum kinetic energy is Kmax = eV₀ (Eq. 11.1).
• Repeat with the same frequency at higher intensities I₂ and I₃ (I₃ > I₂ > I₁). The saturation current rises in proportion to the intensity, but the stopping potential stays exactly the same (Fig. 11.3).
• So, for light of a given frequency, the stopping potential, and with it Kmax, is independent of the intensity. Kmax depends on the light source and the emitter material, not on how bright the light is.
• Exercise 11.3: a cut-off voltage of 1.5 V means Kmax = 1.5 eV = 1.5 × 1.6 × 10⁻¹⁹ J = 2.4 × 10⁻¹⁹ J.

5. Threshold frequency and no time lag

NCERT §11.4.3

• Now keep the intensity the same and change the frequency. The saturation current comes out the same for every frequency, but the stopping potential changes: it is more negative for higher frequencies, V₀₃ > V₀₂ > V₀₁ when ν₃ > ν₂ > ν₁ (Fig. 11.4).
• Higher-frequency light therefore gives photoelectrons a larger maximum kinetic energy, and a larger retarding potential is needed to stop them.
• Plotting stopping potential against frequency gives a straight line for each metal (Fig. 11.5). V₀ varies linearly with ν, and the line meets the frequency axis at a cut-off frequency ν₀ where the stopping potential is zero.
• Two conclusions follow. Kmax rises linearly with frequency but does not depend on intensity. And below ν₀ there is no emission at all, however intense the light. This cut-off ν₀ is the threshold frequency, and it differs from metal to metal.
• Materials respond differently: selenium is more sensitive than zinc or copper. One material also responds differently to different wavelengths: ultraviolet light ejects electrons from copper, but green or red light does not.
• Above the threshold, emission begins at once, with no noticeable delay, even in very dim light. The delay is now known to be of the order of 10⁻⁹ s or less.
• Summary of the observations: (i) the photocurrent is proportional to intensity; (ii) the saturation current is proportional to intensity while the stopping potential is independent of it; (iii) there is a threshold frequency below which nothing is emitted, and above it Kmax rises linearly with frequency but not with intensity; (iv) emission is instantaneous.

6. Where the wave picture fails

NCERT §11.5

• By the end of the nineteenth century the wave theory explained interference, diffraction and polarisation well. In it, light is an electromagnetic wave whose energy is spread continuously over the whole region the wave fills.
• On the wave picture, electrons at the surface soak up energy continuously. Brighter light has larger electric and magnetic field amplitudes, so each electron should gain more energy, and Kmax should rise with intensity. Experiment says it does not.
• Also on the wave picture, a strong enough beam of any frequency, shining long enough, should give electrons enough energy to escape. There should be no threshold frequency. Experiment says there is one.
• The wave spreads its energy over the whole wavefront, shared among a very large number of electrons, so each one gains energy only slowly. Calculations suggest that one electron could need hours or more to collect the work function.
• That clashes with observation (iv): emission is instantaneous. The wave picture fails on observations (i), (ii), (iii) and (iv) alike: it cannot explain the basic features of photoelectric emission.

7. Einstein's photoelectric equation

NCERT §11.6

• In 1905 Einstein proposed that radiation energy comes in discrete units, quanta, each of energy hν, where h is Planck's constant and ν the frequency. Emission is not the slow soaking-up of energy from a wave.
• An electron absorbs one whole quantum hν. If that exceeds the work function φ₀, the electron escapes, and the most it can carry away is Kmax = hν − φ₀ (Eq. 11.2). Electrons bound more tightly come out with less than this.
• Kmax depends linearly on ν and not on intensity, because one electron absorbs one quantum; intensity only fixes how many quanta arrive per unit area per second.
• Since Kmax cannot be negative, emission needs hν > φ₀, that is ν > ν₀ with ν₀ = φ₀/h (Eq. 11.3). A metal with a larger work function has a higher threshold frequency, and below ν₀ nothing comes out, however intense the light or however long it shines.
• More quanta per second means more electrons absorbing them, so for ν > ν₀ the photocurrent is proportional to intensity. And since each absorption is a single, instantaneous event, dim light causes no delay: it only means fewer electrons take part.
• With Kmax = eV₀ the equation becomes eV₀ = hν − φ₀, or V₀ = (h/e)ν − φ₀/e for ν ≥ ν₀ (Eq. 11.4). The V₀–ν graph is a straight line of slope h/e, the same for every material.
• Millikan, between 1906 and 1916, set out to disprove the equation. He measured the slope of the line for sodium and, using the known e, found a value of h close to Planck's constant (6.626 × 10⁻³⁴ J s) obtained in an entirely different context. He ended up confirming the equation for several alkali metals over a wide range of frequencies.
• Example 11.2 (caesium, φ₀ = 2.14 eV): 2.14 eV is 3.42 × 10⁻¹⁹ J, and dividing by h = 6.63 × 10⁻³⁴ J s gives ν₀ = 5.16 × 10¹⁴ Hz. A 0.60 V stopping potential needs hν = 0.60 + 2.14 = 2.74 eV, so λ = hc/hν = (6.63 × 10⁻³⁴ × 3 × 10⁸)/(2.74 × 1.6 × 10⁻¹⁹ J) = 454 nm.
• Exercise 11.5: a V₀–ν slope of 4.12 × 10⁻¹⁵ V s gives h = e × slope = 1.6 × 10⁻¹⁹ × 4.12 × 10⁻¹⁵ = 6.59 × 10⁻³⁴ J s. Exercise 11.6: with ν₀ = 3.3 × 10¹⁴ Hz and ν = 8.2 × 10¹⁴ Hz, V₀ = h(ν − ν₀)/e = 6.63 × 10⁻³⁴ × 4.9 × 10¹⁴/(1.6 × 10⁻¹⁹) = 2.0 V.
• Exercise 11.7: a photon of wavelength 330 nm carries hc/λ = 6.03 × 10⁻¹⁹ J = 3.77 eV, less than a work function of 4.2 eV, so that metal gives no photoelectric emission. Exercise 11.9: 488 nm light carries 2.55 eV; with a stopping potential of 0.38 V the work function is 2.55 − 0.38 = 2.16 eV.

8. The photon

NCERT §11.7

• The photoelectric effect shows that light, when it interacts with matter, acts as if it were made of packets of energy hν.
• Einstein showed that the quantum also carries momentum hν/c. A definite energy together with a definite momentum is a strong sign of a particle, and the particle was later named the photon. Compton's experiment on X-rays scattered by electrons, in 1924, confirmed this particle-like behaviour.
• Einstein received the Nobel Prize in Physics in 1921 for his work in theoretical physics and on the photoelectric effect; Millikan received it in 1923 for his work on the elementary charge and on the photoelectric effect.
• The photon picture: (i) in its interaction with matter, radiation behaves as if made of photons; (ii) each photon has energy E = hν, momentum p = hν/c and speed c.
• (iii) All photons of a given frequency ν, or wavelength λ, have the same energy E = hν = hc/λ and the same momentum p = hν/c = h/λ, whatever the intensity. Brighter light only means more photons crossing a given area per second.
• (iv) Photons carry no charge, so electric and magnetic fields do not deflect them. (v) In a collision between a photon and a particle, total energy and total momentum are conserved, but the number of photons need not be: a photon may be absorbed or a new one created.
• Example 11.1: a laser emits 2.0 × 10⁻³ W of light at 6.0 × 10¹⁴ Hz. Each photon has E = hν = 6.63 × 10⁻³⁴ × 6.0 × 10¹⁴ = 3.98 × 10⁻¹⁹ J, and the number emitted per second is N = P/E = 2.0 × 10⁻³/3.98 × 10⁻¹⁹ = 5.0 × 10¹⁵.
• Exercise 11.1: a 30 kV electron can give at most 30 keV to an X-ray photon, so νmax = eV/h = (30 × 10³ × 1.6 × 10⁻¹⁹)/(6.63 × 10⁻³⁴) = 7.24 × 10¹⁸ Hz, and λmin = c/νmax = 0.0414 nm.

9. Matter waves and de Broglie

NCERT §11.8

• Light shows its wave side in interference, diffraction and polarisation, and its particle side in the photoelectric and Compton effects, which involve transfers of energy and momentum. Which description fits depends on the experiment.
• Seeing uses both. The eye-lens gathers and focuses light as a wave, but the rods and cones of the retina absorb it as photons.
• In 1924 Louis Victor de Broglie proposed that if radiation has two sides, so should matter: moving particles should show wave-like properties in suitable conditions. His argument was that nature is symmetrical between its two basic entities, matter and energy.
• De Broglie relation: a particle of momentum p has wavelength λ = h/p = h/mv (Eq. 11.5), where m is its mass and v its speed. This λ is the de Broglie wavelength, and the wave is a matter wave.
• The relation joins the two sides in one line: λ on the left belongs to a wave, p on the right to a particle, and Planck's constant links them.
• For a material particle the relation is a hypothesis that only experiment can test. It does hold for a photon: with p = hν/c (Eq. 11.6), h/p = c/ν = λ (Eq. 11.7), the ordinary wavelength of the radiation.
• The de Broglie wavelength does not depend on the particle's charge or on what it is made of; only its momentum matters.

10. How long is a matter wave?

NCERT §11.8

• λ = h/p is smaller for a heavier particle (large m) or a faster one (large v).
• A ball of mass 0.12 kg at 20 m s⁻¹ has p = 2.40 kg m s⁻¹ and λ = 6.63 × 10⁻³⁴/2.40 = 2.76 × 10⁻³⁴ m. No instrument can detect a length that small, which is why everyday objects never show wave behaviour.
• Example 11.3(a): an electron (m = 9.11 × 10⁻³¹ kg) at 5.4 × 10⁶ m/s has p = 4.92 × 10⁻²⁴ kg m/s and λ = 6.63 × 10⁻³⁴/4.92 × 10⁻²⁴ = 0.135 nm. That is comparable with X-ray wavelengths.
• Example 11.3(b): a ball of mass 0.150 kg at 30.0 m/s has p = 4.50 kg m/s and λ = 1.47 × 10⁻³⁴ m, about 10⁻¹⁹ times the size of a proton: far beyond measurement.
• In the sub-atomic world the wavelength is significant and measurable: for electrons and protons it is of the order of the spacing of atomic planes in crystals, because their masses, and so their momenta, are so small.
• Exercise 11.10: a 0.040 kg bullet at 1.0 km/s has λ = 1.66 × 10⁻³⁵ m; a 0.060 kg ball at 1.0 m/s has λ = 1.1 × 10⁻³² m; a dust particle of 1.0 × 10⁻⁹ kg at 2.2 m/s has λ = 3.0 × 10⁻²⁵ m. All far too small to observe.
• The wavelength of a matter wave has physical meaning. Its phase velocity does not, but its group velocity does, and it equals the particle's velocity.

Must-know facts

1. Work function φ₀: least energy for an electron to escape a metal surface; 1 eV = 1.602 × 10⁻¹⁹ J.
2. Three ways out: thermionic (heat), field (about 10⁸ V m⁻¹, spark plug), photoelectric (light of suitable frequency).
3. e/m of the electron = 1.76 × 10¹¹ C/kg; elementary charge 1.602 × 10⁻¹⁹ C (Millikan, 1913).
4. Hertz found photoelectric emission in 1887; Hallwachs and Lenard studied it in 1886–1902.
5. Zinc, cadmium and magnesium need ultraviolet; alkali metals (Li, Na, K, Cs, Rb) respond to visible light.
6. Photocurrent and saturation current are proportional to intensity.
7. Stopping potential V₀ is independent of intensity and rises linearly with frequency; Kmax = eV₀.
8. Changing frequency at fixed intensity changes V₀ but leaves the saturation current the same.
9. Below the threshold frequency ν₀ there is no emission, however intense the light.
10. Emission is instantaneous: delay of about 10⁻⁹ s or less.
11. Einstein (1905): Kmax = hν − φ₀; ν₀ = φ₀/h.
12. V₀ = (h/e)ν − φ₀/e: slope h/e, the same for every metal; intercept on the ν axis is ν₀.
13. Millikan (1906–1916) measured the slope for sodium and confirmed the equation; h = 6.626 × 10⁻³⁴ J s.
14. Photon: E = hν = hc/λ, p = hν/c = h/λ, speed c, no charge.
15. More intensity means more photons per second, not more energy per photon.
16. Example 11.1: 2.0 mW at 6.0 × 10¹⁴ Hz → 3.98 × 10⁻¹⁹ J per photon, 5.0 × 10¹⁵ photons per second.
17. Example 11.2: caesium, 2.14 eV → ν₀ = 5.16 × 10¹⁴ Hz; V₀ = 0.60 V → λ = 454 nm.
18. de Broglie (1924): λ = h/p = h/mv.
19. Electron at 5.4 × 10⁶ m/s: λ = 0.135 nm, comparable with X-rays.
20. Ball of 0.150 kg at 30.0 m/s: λ = 1.47 × 10⁻³⁴ m, far beyond measurement.

Common traps

Formulas

Key terms

Work function

The least energy an electron needs to escape from a metal surface, written φ₀.

Electron volt

Energy an electron gains across 1 V: 1.602 × 10⁻¹⁹ J.

Thermionic emission

Electrons escaping from a metal that has been heated strongly.

Field emission

Electrons pulled out of a metal by a very strong electric field.

Photoelectric emission

Electrons ejected from a surface by light of suitable frequency.

Photoelectron

An electron released from a surface by light.

Emitter and collector

The photosensitive plate C that releases electrons, and the plate A that gathers them.

Saturation current

The largest photocurrent, reached when every emitted electron gets to the collector.

Stopping potential

The smallest retarding potential on the collector that brings the photocurrent to zero.

Threshold frequency

The lowest frequency that can eject electrons from a given metal; ν₀ = φ₀/h.

Quantum

A single packet of radiation energy, hν.

Photon

The particle of light: energy hν, momentum h/λ, speed c, no charge.

Planck's constant

h = 6.626 × 10⁻³⁴ J s; links a photon's energy to its frequency.

Dual nature

Radiation and matter each show wave behaviour in some experiments and particle behaviour in others.

Matter wave

The wave associated with a moving material particle.

de Broglie wavelength

λ = h/p, the wavelength of a particle's matter wave.