Photons

Threshold frequency \(f_0\): the minimum frequency of the incident radiation that can just eject photoelectrons from a given metal surface. Below it no emission occurs, however intense the light.

Stopping potential \(V_s\): the minimum retarding (negative) potential of the anode with respect to the cathode that stops even the most energetic photoelectrons, reducing the photocurrent to zero.

Work function \(\phi\): the minimum energy needed by an electron to just escape from the metal surface; \(\phi=hf_0\).

In the classical wave theory the energy is spread continuously over the wavefront, so a low-frequency but intense (or long-lasting) wave should eventually give an electron enough accumulated energy to escape. The wave theory therefore predicts no threshold frequency.

In the quantum theory light consists of photons of energy \(E=hf\), and one electron absorbs one photon. If \(hf<\phi\) the electron cannot escape, no matter how many photons arrive per second or for how long. The observed sharp threshold frequency fits this one-photon, one-electron interaction.

The radiation arrives as photons of energy \(E=hf\). An electron needs at least the work function \(\phi\) to leave the metal. If \(f\) is below the threshold, \(hf<\phi\), and since an electron absorbs only one photon at a time it never gets enough energy to escape.

Instantaneous emission: electrons are emitted almost at once (in less than about \(10^{-9}\ \mathrm{s}\)) when light strikes the surface, with none of the time delay for energy to build up that wave theory predicts.

Kinetic energy independent of intensity: the maximum kinetic energy of the electrons depends only on the frequency of the light, not on its intensity.

Solar (photovoltaic) cells, which convert sunlight directly into electrical energy; and light meters in photography, which measure light intensity. (Automatic doors and burglar alarms also use photocells.)

Millikan used a vacuum tube with an alkali-metal cathode and an anode. Monochromatic light of several known frequencies \(f\) was shone on the cathode. For each frequency a variable retarding voltage was applied to the anode until the photocurrent just became zero, giving the stopping potential \(V_s\).

From Einstein’s equation, \(eV_s=hf-\phi\), so

A graph of \(V_s\) against \(f\) is a straight line of slope \(\dfrac he\). Multiplying the measured slope by the electron charge \(e\) gives Planck’s constant \(h\).

(ii) Kinetic energy:

(i) Stopping potential:

(iii) Velocity:

Photoelectric effect: the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency falls on it.

Einstein’s equation: light consists of photons of energy \(E=hf\). A photon gives its whole energy to a single electron. Part of it, the work function \(\phi\), is used to overcome the surface barrier, and the rest appears as the maximum kinetic energy of the emitted electron. By conservation of energy

Laws from the equation:

Threshold frequency: emission needs \(K_{\max}\ge0\), that is \(hf\ge\phi\), so there is a minimum frequency \(f_0=\phi/h\) below which no electrons are emitted.

Independence of intensity: \(K_{\max}\) depends only on \(f\) and \(\phi\). A greater intensity means more photons per second, hence a larger photocurrent, but not a larger \(K_{\max}\).

Dependence on frequency: \(K_{\max}\) increases linearly with the frequency of the light.

Using \(hc\approx1240\ \mathrm{eV\,nm}\), with \(\lambda=181\ \mathrm{nm}\) and \(\lambda_0=264\ \mathrm{nm}\):

Einstein postulated that light interacts with matter in quanta. An electron near the surface absorbs one whole photon of energy \(hf\). If \(hf>\phi\), the electron uses \(\phi\) to break free of the binding forces and keeps the excess as kinetic energy:

This is Einstein’s photoelectric equation (see Question 4a).

Photosynthesis is a photochemical reaction, and each molecule that starts the reaction must absorb a single photon with at least a certain minimum energy. Sunlight contains visible and ultraviolet light, whose photons have a high frequency and energy \(hf\) large enough to trigger the reaction. Infrared photons have a lower frequency and so too little energy per photon. Since a molecule absorbs one photon at a time, the infrared photons cannot start the process, however intense the infrared light is.

(a) Photons per second: the power through the area is

The energy of one photon is

(b) Momentum of a photon:

(c) Force: the light is totally absorbed, so each photon loses its whole momentum \(p\) and the force is the rate of change of momentum:

The photoelectric effect.

(i) Threshold frequency: from \(E_k=hf-\phi\), \(E_k=0\) at \(f=f_0\), so \(f_0\) is the intercept of the graph on the frequency axis. From the figure, \(f_0=6.8\times10^{14}\ \mathrm{Hz}\).

(ii) Planck’s constant: the gradient of the straight line is \(h\). Taking the gradient of the line in the figure,

A lower work function means a lower threshold frequency, \(f_0=\phi/h\). The gradient is still \(h\), so the new line must be a straight line parallel to the original, shifted to the left, so that it meets the frequency axis at a smaller value of \(f\).

Einstein’s equation gives the energy of electrons that come from the very surface of the metal and lose nothing on the way out. Electrons from deeper in the metal need more than \(\phi\) to reach the surface, because they lose varying amounts of energy in collisions with atoms. They therefore leave with a range of kinetic energies from zero up to the maximum value.

A photon is a discrete packet, or quantum, of electromagnetic radiation energy.

(i) Wavelength:

(ii) Momentum:

The photoelectric effect is the ejection of electrons from the surface of a material, usually a metal, when electromagnetic radiation of sufficiently high frequency falls on it.

Use \(\phi=\dfrac{hc}{\lambda_0}\), so \(\lambda_0=\dfrac{hc}{\phi}\) with \(hc\approx1240\ \mathrm{eV\,nm}\):

Emission occurs only if the incident wavelength is less than or equal to the threshold wavelength, that is, if the photon energy is at least the work function.

Caesium: \(436\ \mathrm{nm}<886\ \mathrm{nm}\), so the photon energy (about 2.8 eV) exceeds 1.4 eV and photoelectric emission occurs.

Tungsten: \(436\ \mathrm{nm}>276\ \mathrm{nm}\), so the photon energy (about 2.8 eV) is less than 4.5 eV and there is no photoelectric emission.