Concept of quantization of energy

PHOTO ELECTRIC EFFECT

Concept of quantization of energy

Failures of classical wave theory

From Maxwell’s theory (Refer unit 5 of volume 1), we learnt that light is an electromagnetic wave consisting of coupled electric and magnetic oscillations that move with the speed of light and exhibit typical wave behaviour. Let us try to explain the experimental observations of photoelectric effect using wave picture of light.

i) When light is incident on the target, there is a continuous supply of energy to the electrons. According to wave theory, light of greater intensity should impart greater kinetic energy to the liberated electrons (Here, Intensity of light is the energy delivered per unit area per unit time).

But this does not happen. The experiments show that maximum kinetic energy of the photoelectrons does not depend on the intensity of the incident light.

ii) According to wave theory, if a sufficiently intense beam of light is incident on the surface, electrons will be liberated from the surface of the target, however low the frequency of the radiation is.

From the experiments, we know that photoelectric emission is not possible below a certain minimum frequency. Therefore, the wave theory fails to explain the existence of threshold frequency.

iii) Since the energy of light is spread across the wavefront, the electrons which receive energy from it are large in number. Each electron needs considerable amount of time (a few hours) to get energy sufficient to overcome the work function and to get liberated from the surface.

But experiments show that photoelectric emission is almost instantaneous process (the time lag is less than 10^–9 s after the surface is illuminated) which could not be explained by wave theory.

Thus, the experimental observations of photoelectric emission could not be explained on the basis of the wave theory of light.

EXAMPLE 7.1

For the photoelectric emission from cesium, show that wave theory predicts that

i) maximum kinetic energy of the photoelectrons (K_max) depends on the intensity I of the incident light

ii) K_max does not depend on the frequency of the incident light and

iii) the time interval between the incidence of light and the ejection of photoelectrons is very long.

(Given : The work function for cesium is 2.14 eV and the power absorbed per unit area is 1.60×10^−6Wm^−2 which produces a measurable photocurrent in cesium.)

Solution

i) According to wave theory, the energy in a light wave is spread out uniformly and continuously over the wavefront. For the sake of simplicity, the following assumptions are made.

a) Light is absorbed in the top atomic layer of the metal

b) For a given element, each atom absorbs an equal amount of energy and this energy is proportional to its cross-sectional area A

c) Each atom gives this energy to one of the electrons.

The energy absorbed by each electron in time t is given by

E = IAt

With this energy absorbed, the most energetic electron is released with Kmax by overcoming the surface energy barrier or work function Ï•₀ and this is expressed as

K_max = IAt − Ï•₀           (1)

Thus, wave theory predicts that for a unit time, at low light intensities when IA < Ï•₀, no electrons are emitted. At higher intensities, when IA ≥ Ï•₀, electrons are emitted. This implies that higher the light intensity, greater will be K_max.

Therefore, the predictions of wave theory contradict experimental observations at both very low and very high light intensities.

K_max is dependent only on the intensity under given conditions – that is, by suitably increasing the intensity, one can produce  photoelectric effect even if the frequency is less than the threshold frequency. So the concept of threshold frequency does not even exist in wave theory.

ii) According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field (E₀²). The amplitude of this electric field increases with increasing intensity and imparts an increasing acceleration and kinetic energy to an electron.

Now I is replaced with a quantity proportional to E² in equation (1). This means that K_max should not depend at all on the frequency of the classical light wave which again contradicts the experimental results.

iii) If an electron accumulates light energy just enough to overcome the work function, then it is ejected out of the atom with zero kinetic energy. Therefore, from equation (1),

By taking the atomic radius r = 1.0×10^−10 m and substituting the given values of I and Ï•₀, we can estimate the time interval as

t = 2.14×1.6×10^−19 / 1.60×10^−6 ×3.14×(1×10^−10)²

= 0.68×10⁷ s ≈ 79 days

Thus, wave theory predicts that there is a large time gap between the incidence of light and the ejection of photoelectrons but the experiments show that photo emission is an instantaneous process.

Concept of quantization of energy

Max Planck proposed quantum concept in 1900 in order to explain the thermal radiations emitted by a black body and the shape of its radiation curves.

According to Planck, matter is composed of a large number of oscillating particles (atoms) which vibrate with different frequencies. Each atomic oscillator - which vibrates with its characteristic frequency - emits or absorbs electromagnetic radiation of the same frequency. It also says that

i) If an oscillator vibrates with frequency v, its energy can have only certain discrete values, given by the equation.

E_n= nhν       n=1,2,3....     (7.4)

where h is a constant, called Planck’s constant.

ii) The oscillators emit or absorb energy in small packets or quanta and the energy of each quantum is E = hν.

This implies that the energy of the oscillator is quantized – that is, energy is not continuous as believed in the wave picture. This is called quantization of energy.