11.9 Summary

• The minimum energy needed by an electron to come out from a metal surface is called the work function of the metal. Energy (greater than the work function \(\left(\phi_0\right)\) required for electron emission from the metal surface can be supplied by suitably heating or applying strong electric field or irradiating it by light of suitable frequency.
• The photoelectric effect is the phenomenon of emission of electrons by metals when illuminated by light of suitable frequency. Certain metals respond to ultraviolet light while others are sensitive even to the visible light. The photoelectric effect involves conversion of light energy into electrical energy. It follows the law of conservation of energy. The photoelectric emission is an instantaneous process and possesses certain special features.
• Photoelectric current depends on (i) the intensity of incident light, (ii) the potential difference applied between the two electrodes, and (iii) the nature of the emitter material.
• The stopping potential \(\left(V_o\right)\) depends on (i) the frequency of incident light, and (i1) the nature of the emitter material. For a given frequency of incident light, it is independent of its intensity. The stopping potential is directly related to the maximum kinetic energy of electrons emitted: \(e V_0=(1 / 2) m v_{\max }^2=K_{\max }\).
• Below a certain frequency (threshold frequency) \(\nu_0\), characteristic of the metal, no photoelectric emission takes place, no matter how large the intensity may be.
• The classical wave theory could not explain the main features of the photoelectric effect. Its picture of continuous absorption of energy from radiation could not explain the independence of \(K_{\max }\) on intensity, the existence of \(\nu_{\mathrm{o}}\) and the instantaneous nature of the process. Einstein explained these features on the basis of photon picture of light. According to this, light is composed of discrete packets of energy called quanta or photons. Each photon carries an energy \(E(=h \nu)\) and momentum \(p(=h / \lambda)\), which depend on the frequency ( \(\nu\) ) of incident light and not on its intensity. Photoelectric emission from the metal surface occurs due to the absorption of a photon by an electron.
• Einstein’s photoelectric equation is in accordance with the energy conservation law as applied to the photon absorption by an electron in the metal. The maximum kinetic energy \((1 / 2) m v_{\max }^2\) is equal to the photon energy ( \(h \nu\) ) minus the work function \(\phi_0\left(=h \nu_o\right)\) of the target metal:
  \(
  \frac{1}{2} m v_{\max }^2=V_0 e=h \nu-\phi_0=h\left(\nu-\nu_0\right)
  \)
• This photoelectric equation explains all the features of the photoelectric effect. Millikan’s first precise measurements confirmed the Einstein’s photoelectric equation and obtained an accurate value of Planck’s constant \(h\). This led to the acceptance of the particle or photon description (nature) of electromagnetic radiation, introduced by Einstein.
• Radiation has dual nature: wave and particle. The nature of the experiment determines whether a wave or particle description is best suited for understanding the experimental result. Reasoning that radiation and matter should be symmetrical in nature, Louis Victor de Broglie attributed a wave-like character to matter (material particles). The waves associated with the moving material particles are called matter waves or de Broglie waves.
• The de Broglie wavelength \((\lambda)\) associated with a moving particle is related to its momentum \(p\) as: \(\lambda=h / p\). The dualism of matter is inherent in the de Broglie relation which contains a wave concept \((\lambda)\) and a particle concept ( \(p\) ). The de Broglie wavelength is independent of the charge and nature of the material particle. It is significantly measurable (of the order of the atomic-planes spacing in crystals) only in the case of sub-atomic particles like electrons, protons, etc. (due to the smallness of their masses and hence, momenta). However, it is indeed very small, quite beyond measurement, in the case of macroscopic objects, commonly encountered in everyday life.

POINTS TO PONDER

• Free electrons in a metal are free in the sense that they move inside the metal in a constant potential (This is only an approximation). They are not free to move out of the metal. They need additional energy to get out of the metal.
• Free electrons in a metal do not all have the same energy. Like molecules in a gas jar, the electrons have a certain energy distribution at a given temperature. This distribution is different from the usual Maxwell’s distribution that you have learnt in the study of the kinetic theory of gases. You will learn about it in later courses, but the difference has to do with the fact that electrons obey Pauli’s exclusion principle.
• Because of the energy distribution of free electrons in a metal, the energy required by an electron to come out of the metal is different for different electrons. Electrons with higher energy require less additional energy to come out of the metal than those with lower energies. Work function is the least energy required by an electron to come out of the metal.
• Observations on the photoelectric effect imply that in the event of matterlight interaction, absorption of energy takes place in discrete units of \(h \nu\). This is not quite the same as saying that light consists of particles, each of energy \(h \nu\).
• Observations on the stopping potential (its independence of intensity and dependence on frequency) are the crucial discriminator between the wave-picture and photon-picture of the photoelectric effect.
• The wavelength of a matter wave given by \(\lambda=\frac{h}{p}\) has physical significance; its phase velocity \(v_p\) has no physical significance. However, the group velocity of the matter wave is physically meaningful and equals the velocity of the particle.