11.6 Einstein’s Photoelectric Equation: Energy Quantum of Radiation

In 1905, Albert Einstein (1879-1955) proposed a radically new picture of electromagnetic radiation to explain photoelectric effect. In this picture, photoelectric emission does not take place by continuous absorption of energy from radiation. Radiation energy is built up of discrete units – the so called quanta of energy of radiation. Each quantum of radiant energy has energy \(h \nu\), where \(h\) is Planck’s constant and \(\nu\) the frequency of light. In photoelectric effect, an electron absorbs a quantum of energy ( \(h \nu\) ) of radiation. If this quantum of energy absorbed exceeds the minimum energy needed for the electron to escape from the metal surface (work function \(\phi_0\) ), the electron is emitted with maximum kinetic energy
\(
K_{\max }=h \nu-\phi_0 \dots(11.2)
\)
More tightly bound electrons will emerge with kinetic energies less than the maximum value. Note that the intensity of light of a given frequency is determined by the number of photons incident per second. Increasing the intensity will increase the number of emitted electrons per second. However, the maximum kinetic energy of the emitted photoelectrons is determined by the energy of each photon.

Equation (11.2) is known as Einstein’s photoelectric equation. We now see how this equation accounts in a simple and elegant manner all the observations on the photoelectric effect given at the end of sub-section 11.4.3.

• According to Eq. (11.2), \(K_{\max }\) depends linearly on \(\nu\), and is independent of the intensity of radiation, in agreement with observation. This has happened because in Einstein’s picture, the photoelectric effect arises from the absorption of a single quantum of radiation by a single electron. The intensity of the radiation (that is proportional to the number of energy quanta per unit area per unit time) is irrelevant to this basic process.
• Since \(K_{\max }\) must be non-negative, Eq. (11.2) implies that photoelectric emission is possible only if
  \(
  h \nu>\phi_0
  \)
  or \(\nu>\nu_0\), where
  \(
  \nu_0=\frac{\phi_0}{h} \dots(11.3)
  \)
  Equation (11.3) shows that the greater the work function \(\phi_0\), the higher the minimum or threshold frequency \(\nu_0\) needed to emit photoelectrons. Thus, there exists a threshold frequency \(\nu_0\left(=\phi_0 / h\right)\) for the metal surface, below which no photoelectric emission is possible, no matter how intense the incident radiation may be or how long it falls on the surface.
• In this picture, the intensity of radiation as noted above, is proportional to the number of energy quanta per unit area per unit time. The greater the number of energy quanta available, the greater is the number of electrons absorbing the energy quanta and greater, therefore, is the number of electrons coming out of the metal (for \(\nu>\nu_0\) ). This explains why, for \(\nu>\nu_0\), photoelectric current is proportional to intensity.
• In Einstein’s picture, the basic elementary process involved in photoelectric effect is the absorption of a light quantum by an electron. This process is instantaneous. Thus, whatever may be the intensity i.e., the number of quanta of radiation per unit area per unit time, photoelectric emission is instantaneous. Low intensity does not mean delay in emission, since the basic elementary process is the same. Intensity only determines how many electrons are able to participate in the elementary process (absorption of a light quantum by a single electron) and, therefore, the photoelectric current.

Using Eq. (11.1), the photoelectric equation, Eq. (11.2), can be written as
\(
e V_0=h \nu-\phi_0 ; \text { for } \nu \geq \nu_0
\)
\(
\text { or } V_0=\frac{h}{e} \nu-\frac{\phi_0}{e} \dots(11.4)
\)
This is an important result. It predicts that the \(V_0\) versus \(\nu\) curve is a straight line with slope \(=(h / e)\), independent of the nature of the material. During 1906-1916, Millikan performed a series of experiments on the photoelectric effect, aimed at disproving Einstein’s photoelectric equation. He measured the slope of the straight line obtained for sodium, similar to that shown in Fig. 11.5. Using the known value of \(e\), he determined the value of Planck’s constant \(h\). This value was close to the value of Planck’s constant \(\left(=6.626 \times 10^{-}\right.\) \({ }^{34} \mathrm{~J}\) s) determined in an entirely different context. In this way, in 1916, Millikan proved the validity of Einstein’s photoelectric equation, instead of disproving it.

The successful explanation of the photoelectric effect using the hypothesis of light quanta and the experimental determination of values of \(h\) and \(\phi_0\), in agreement with values obtained from other experiments, led to the acceptance of Einstein’s picture of the photoelectric effect. Millikan verified the photoelectric equation with great precision, for a number of alkali metals over a wide range of radiation frequencies.