Physics » Optics and Optical Phenomena » The Photoelectric Effect

# Implications of Einstein’s Model

## Implications of Einstein’s Model

For a paper on the history of the photoelectric effect and the analysis of its treatmentin a variety of textbooks see: http://onlinelibrary.wiley.com/doi/10.1002/sce.20389/pdf

Einstein’s model is consistent with the observation that the electrons wereemitted immediately when light was shone on the metal and that the intensity of the light made no difference to the maximum kinetic energy of the emitted electrons.

The energy needed to knock an electron out of the substance is called the work function (symbol $$W_{0}$$) of the substance. This is a characteristic of the substance. If the energy of the photon is less than the work function then no electron can be emitted, no matter how many photons strike the substance. We know that the frequency of light is related to the energy, that is why there is a minimum frequency of light that can eject electrons. This minimum frequency we call the cut-off frequency, $$f_0$$. For a specific colour of light (i.e. a certain frequency or wavelength), the energy of the photons is given by $$E=hf=\frac{hc}{\lambda}$$, where $$h$$ is Planck’s constant. This tells us that the $$W_{0} = hf_0$$.

### Definition: Work function

The minimum energy needed to knock an electron out of a metal is called the work function (symbol $$W_{0}$$) of the metal. As it is energy, it measured in joules (J).

Energy is conserved so if the photon has a higher energy than $$W_{0}$$then the excess energy goes into the kinetic energy $${E}_{k}$$ of the electronthat was emitted fromthe substance.

The excess over and above the binding energy is actually the maximum kinetic energy the emitted electron canhave. This is because not all electrons are on the surface of the substance. For electrons below the surface there is additional energy required to eject the electron from the material which then cannot contribute to the kinetic energy of the electron.

\begin{align*} E & = W_{0}+{E}_{k\max} \\ {E}_{k\max}& = hf-W_{0} \end{align*}

This equation is known as the photoelectric equation.

The last piece of the puzzle is now clear, the question was `why does increasing theintensity of the light not affect the maximum kinetic energy of the emitted photons?’. The answeris that each emitted electron has absorbed one photon, increasingthe intensity just increases the number of photons (we expect more electrons but we don’t expecttheir maximum kinetic energy to change).

The discovery and understanding of the photoelectric effect was one of the major breakthroughs in science in the twentieth century as it provided concrete evidence of the particle nature of light. It overturned previously held views that light was composed purely of a continuous transverse wave. On the one hand, the wave nature is a good description of phenomena such as diffraction and interference for light, and on the other hand, the photoelectric effect demonstrates the particle nature of light. This is now known as the ‘dual-nature’ of light. (dual means two)

Einstein won the 1921 Nobel Prize for Physics for this quantum theory and his explanation of the photoelectric effect. The photoelectric effect: Incoming photons on the left hit the electrons inside the metal surface. The electrons absorb the energy from the photons, and are ejected from the metal surface.

We can observe this effect in the following practical demonstration of photoelectric emission. A zinc plate is charged negatively and placed onto the cap of an electroscope. In the figure below, red light is shone onto the zinc plate. There is no change observed even if the intensity (brightness) of the red light is increased. In the second figure below, ultraviolet light of low intensity is shone onto the zinc and it is observed that the leaf of the electroscope collapse. This allows us to conclude that the negative charge on the plate decreased as electrons were ejected from the metal when the ultraviolet light was incident on the plate. Red light incident on the zinc plate of an electroscope. Ultraviolet light incident on the zinc plate of an electroscope.

### Optional Video: The Wave/Particle Duality – Part 1

The work function is different for different elements. The smaller the work function, the easier it is for electrons to be emitted from the metal. Metals with low work functions make good conductors. This is because the electrons are attached less strongly to their surroundings and can move more easily through these materials. This reduces the resistance of the material to the flow of current i.e. it conducts well. The table below shows the work functions for a range of elements.

 Element Work Function (J) Aluminium $$\text{6.9} \times \text{10}^{-\text{19}}$$ Beryllium $$\text{8.0} \times \text{10}^{-\text{19}}$$ Calcium $$\text{4.6} \times \text{10}^{-\text{19}}$$ Copper $$\text{7.5} \times \text{10}^{-\text{19}}$$ Gold $$\text{8.2} \times \text{10}^{-\text{19}}$$ Lead $$\text{6.9} \times \text{10}^{-\text{19}}$$ Silicon $$\text{1.8} \times \text{10}^{-\text{19}}$$ Silver $$\text{6.9} \times \text{10}^{-\text{19}}$$ Sodium $$\text{3.7} \times \text{10}^{-\text{19}}$$

Table: Work functions of selected elements determined from the photoelectric effect. (From the Handbook of Chemistry and Physics.)