6.2 Photoelectric effect  (Page 4/17)

The existence of the cut-off frequency $f_c$ for the photoelectric effect follows from [link] because the kinetic energy $K_{\text{max}}$ of the photoelectron can take only positive values. This means that there must be some threshold frequency for which the kinetic energy is zero, $0=h{f}_c-\varphi .$ In this way, we obtain the explicit formula for cut-off frequency:

Cut-off frequency depends only on the work function of the metal and is in direct proportion to it. When the work function is large (when electrons are bound fast to the metal surface), the energy of the threshold photon must be large to produce a photoelectron, and then the corresponding threshold frequency is large. Photons with frequencies larger than the threshold frequency $f_c$ always produce photoelectrons because they have $K_{\text{max}}>0.$ Photons with frequencies smaller than $f_c$ do not have enough energy to produce photoelectrons. Therefore, when incident radiation has a frequency below the cut-off frequency, the photoelectric effect is not observed. Because frequency f and wavelength $\lambda$ of electromagnetic waves are related by the fundamental relation $\lambda f=c$ (where $c$ is the speed of light in vacuum), the cut-off frequency has its corresponding cut-off wavelength     ${\lambda }_c:$

In this equation, $hc=1240\text{eV}·\text{nm}.$ Our observations can be restated in the following equivalent way: When the incident radiation has wavelengths longer than the cut-off wavelength, the photoelectric effect does not occur.

[link] in Einstein’s model tells us that the maximum kinetic energy of photoelectrons is a linear function of the frequency of incident radiation, which is illustrated in [link] . For any metal, the slope of this plot has a value of Planck’s constant. The intercept with the $K_{\text{max}}$ -axis gives us a value of the work function that is characteristic for the metal. On the other hand, $K_{\text{max}}$ can be directly measured in the experiment by measuring the value of the stopping potential $\text{Δ}{V}_s$ (see [link] ) at which the photocurrent stops. These direct measurements allow us to determine experimentally the value of Planck’s constant, as well as work functions of materials.

Einstein’s model also gives a straightforward explanation for the photocurrent values shown in [link] . For example, doubling the intensity of radiation translates to doubling the number of photons that strike the surface per unit time. The larger the number of photons, the larger is the number of photoelectrons, which leads to a larger photocurrent in the circuit. This is how radiation intensity affects the photocurrent. The photocurrent must reach a plateau at some value of potential difference because, in unit time, the number of photoelectrons is equal to the number of incident photons and the number of incident photons does not depend on the applied potential difference at all, but only on the intensity of incident radiation. The stopping potential does not change with the radiation intensity because the kinetic energy of photoelectrons (see [link] ) does not depend on the radiation intensity.