Dual Nature of Radiation and Matter

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 11.1 to 11.11.

The Davisson-Germer experiment has been removed. Searching this chapter returns zero hits for "Davisson" and zero for "Germer". Older editions closed with a full section describing how electron diffraction from a nickel crystal confirmed the de Broglie hypothesis experimentally.

This leaves the chapter in an unusual position: it asserts that matter has wave properties and gives the formula, but the experiment that proved it is gone. Section 11.8 still shows the de Broglie relation and applies it, so the formula is examinable; the confirming experiment is not.

The photocell has been removed — zero hits. Its applications, from automatic door openers to light meters, no longer appear.

Heisenberg's uncertainty principle has been removed — zero hits for both "uncertainty" and "Heisenberg". Older editions carried it as a boxed discussion alongside the wave nature of matter.

What remains is well supported. Hallwachs and Lenard's observations, Millikan's measurement, the work function, the three experimental laws, Einstein's equation and the photon picture are all present and fully developed.

2. Electron Emission and the Photoelectric Effect (Textbook 11.2 to 11.4)

Electrons are held inside a metal by an energy barrier. The minimum energy needed to free one from the surface is the work function ${\varphi }_0$, typically a few electronvolts and measured in eV for that reason.

Three ways of supplying it, of which only the last concerns this chapter: heating the metal (thermionic emission), applying a strong external field (field emission), and shining light on it (photoelectric emission).

The three experimental laws (11.4), established by Hallwachs, Lenard and Millikan:

• Threshold frequency. Below a certain frequency ${\nu }_0$, characteristic of the metal, no electrons are emitted however intense the light or however long one waits.
• Intensity controls number, not energy. Above threshold, raising the intensity increases the photocurrent but leaves the maximum kinetic energy unchanged.
• Frequency controls energy. The maximum kinetic energy rises linearly with frequency and does not depend on intensity at all.

Emission is also instantaneous, within about $10^{-9}$ s.

The stopping potential is the reverse voltage that just halts the fastest electrons:

$e{V}_0=K_{\max }$

Measured in volts it is numerically equal to $K_{\max }$ in electronvolts, which is what makes the electronvolt so convenient here.

3. Why the Wave Theory Fails (Textbook 11.5)

Every one of the three laws contradicts the classical wave picture, and this section is a standard three-mark question.

The reasoning behind the time lag. On the wave picture, energy arrives spread continuously over the whole surface, so a single electron must wait to accumulate enough. For a dim source the calculated wait runs to hours. No such delay is ever observed.

Einstein's resolution (11.6). Light is absorbed not continuously but in discrete quanta of energy $h\nu$, one photon to one electron. The electron spends ${\varphi }_0$ escaping and keeps the rest:

$K_{\max }=h\nu -{\varphi }_0$

Every law now follows in one line. If $h\nu <{\varphi }_0$ nothing is emitted, whatever the intensity — that is the threshold. Intensity means the number of photons, so it fixes how many electrons come out, not their energy. And absorption is a single event, so there is no delay.

Rewriting as a straight line. Dividing by $e$:

$V_0=\left(\frac{h}{e}\right)\nu -\frac{{\varphi }_0}{e}$

A graph of stopping potential against frequency is a straight line of slope $h/e$ — and crucially that slope contains no material constant, so it is the same for every metal. Only the intercept changes. Millikan's measurement of that slope confirmed Einstein's equation and gave an independent value of $h$, which is Exercise 11.5.

4. The Photon (Textbook 11.7)

Radiation of frequency $\nu$ behaves as a stream of particles with:

$E=h\nu =\frac{hc}{\lambda },p=\frac{h\nu }{c}=\frac{h}{\lambda }$

Photons carry momentum without having mass. Rest mass is zero, so $p=mv$ does not apply; for a massless particle $p=E/c$ instead. Photons are electrically neutral, travel at $c$ in vacuum, and are unaffected by electric or magnetic fields.

A photon count is enormous. Exercise 11.4 finds a 9.42 mW laser emitting $3\times 10^{16}$ photons per second, which is why a beam looks perfectly smooth rather than granular.

The X-ray limit. Running the photoelectric logic backwards, an electron accelerated through $V$ produces at most a photon of energy $eV$:

$h{\nu }_{\max }=eV$

giving a sharp short-wavelength cut-off — Exercise 11.1. Classical theory predicts no such limit.

5. Matter Waves (Textbook 11.8)

De Broglie's proposal was that the duality runs both ways: if waves behave as particles, particles should behave as waves, with:

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

For an accelerated electron, substituting $p=\sqrt{2mK}$ with $K=eV$:

$\lambda =\frac{h}{\sqrt{2meV}}$

which gives about 1.23 nm$/\sqrt{V}$ — comparable to atomic spacings for modest voltages, which is exactly why electrons diffract from crystals.

Why everyday objects show nothing. Exercise 11.10 computes wavelengths of $10^{-35}$ to $10^{-25}$ m for a bullet, a ball and a dust particle. Even the largest is ten orders of magnitude smaller than a nucleus, so no aperture could ever diffract them. The wavelength is inversely proportional to mass, so only very light particles show wave behaviour.

The consistency check. Applying the de Broglie relation to a photon returns the ordinary wavelength of the radiation, since ${\lambda }_{dB}=h/(h\nu /c)=c/\nu =\lambda$. That is Exercise 11.11, and it shows the wave and particle descriptions are one coherent scheme rather than two rival ones.

Summary

• The work function ${\varphi }_0$ is the minimum energy needed to free an electron from a metal surface, of order a few eV.
• Emission can be thermionic, field-induced or photoelectric; only the last is treated here.
• Threshold frequency: below ${\nu }_0$ there is no emission at any intensity or duration.
• Intensity sets the number of photoelectrons; frequency sets their maximum energy.
• Emission is effectively instantaneous, under $10^{-9}$ s.
• Stopping potential: $e{V}_0=K_{\max }$, so $V_0$ in volts equals $K_{\max }$ in electronvolts.
• Wave theory fails on all three counts: it predicts no threshold, energy rising with intensity, and a long time lag.
• Einstein's equation $K_{\max }=h\nu -{\varphi }_0$ follows from one photon being absorbed by one electron.
• $V_0$ against $\nu$ is a straight line of slope $h/e$ — the same for every metal, with only the intercept changing.
• Photon: $E=h\nu =hc/\lambda$ and $p=h\nu /c=h/\lambda$.
• Photons have zero rest mass but non-zero momentum, since $p=E/c$ for a massless particle.
• They are neutral, travel at $c$, and are undeflected by electric and magnetic fields.
• X-ray cut-off: $h{\nu }_{\max }=eV$, giving a minimum wavelength classical theory cannot explain.
• De Broglie: $\lambda =h/mv$, and $\lambda =h/\sqrt{2meV}$ for an electron accelerated through $V$.
• Matter wavelengths are inversely proportional to mass, so everyday objects show none.
• Applying the de Broglie relation to a photon returns $c/\nu$, confirming the two pictures agree.
• The Davisson-Germer experiment, the photocell and Heisenberg's uncertainty principle have all been removed from this chapter.