1.2 Einstein’s Photon Explanation
Einstein made the radical postulate that a beam of light consists of small packages of energy called photons or quanta. This postulate was an extension of an idea developed five years earlier by Max Planck to explain the properties of blackbody radiation. In Einstein’s picture, the energy E of an individual photon is equal to a constant h times the photon frequency ƒ. From the relationship for electromagnetic waves in vacuum, we have
E = hf
=
(9.2)
where h is a universal constant called Planck’s constant. The numerical value of this constant, to the accuracy known at present, is
h = 6,62606896·10-34 J·s
CAUTION Photons are not “particles” in the usual sense It’s common to envision photons as miniature billiard balls or pellets. While that’s a convenient mental picture, it’s not very accurate. For one thing, billiard balls and bullets have a rest mass and travel slower than the speed of light c, while photons travel at the speed of light and have zero rest mass. For another thing, photons have wave aspects (frequency and wavelength) that are easy to observe. The fact is that the photon concept is a very strange one, and the true nature of photons is difficult to visualize in a simple way.
In Einstein’s picture, an individual photon arriving at the surface in Fig. 9.1. This energy transfer is an all-or-nothing process, in contrast to the continuous transfer of energy in the wave theory of light; the electron gets all of the photon’s energy or none at all. The electron can escape from the surface only if the energy it acquires is greater than the work function ɸ. Thus photoelectrons will be ejected only if hf > ɸ , or f > ɸ/h.
Einstein’s postulate therefore explains why the photoelectric effect occurs only for frequencies greater than a minimum threshold frequency. This postulate is also consistent with the observation that greater intensity causes a greater photocurrent (Fig. 9.4). Greater intensity at a particular frequency means a greater number of photons per second absorbed, and thus a greater number of electrons emitted per second and a greater photocurrent.
Einstein’s postulate also explains why there is no delay between illumination and the emission of photoelectrons. As soon as photons of sufficient energy strike the surface, electrons can absorb them and be ejected.
Finally, Einstein’s postulate explains why the stopping potential for a given surface depends only on the light frequency. Recall that ɸ is the minimum energy needed to remove an electron from the surface. Einstein applied conservation of energy to find that the maximum kinetic energy for an emitted electron is the energy gained from a photon minus the work function
Kmax
=
–
ɸ (9.3)
Substituting from Kmax = eV0 from Eq. (38.1), we find
eV0 = hf – ɸ (9.4)
Equation (9.4) shows that the stopping potential V0 increases with increasing frequency ƒ. The intensity doesn’t appear in Eq. (9.4), so V0 is independent of intensity. As a check of Eq. (9.4), we can measure the stopping potential for each of several values of frequency for a given cathode material (Fig. 9.5). A graph of V0 as a function of f turns out to be a straight line, verifying Eq. (9.4), and from such a graph we can determine both the work function ɸ for the material and the value of the quantity h/e . After the electron charge –e was measured by Robert Millikan in 1909, Planck’s constant h could also be determined from these measurements.
Electron energies and work functions are usually expressed in electron volts (eV). To four significant figures,
1eV = 1,602·10-19 J
To this accuracy, Planck’s constant is
h = 6,626·10-34J·s = 4,136·10-15eV·s
Table 9.1 Work Functions of Several Elements |
Figure 9,6. Stopping potential as a function of frequency for two cathode materials having different work functions . f. |
Table 9.1
lists the work functions of several elements. These values are
approximate because they are very sensitive to surface impurities.
The greater the work function, the higher the minimum frequency
needed to emit photoelectrons (Fig. 9.6). Stopping potential is zero
at threshold frequency (electrons emerge with zero kinetic energy).
For each material, eV
= hf
ɸ
or V0
=
,
so
the plots have same slope h/e
but different intercepts
on the vertical axis.
The photon picture explains a number of other phenomena in which light is absorbed. One example is a suntan, which is caused when the energy in sunlight triggers a chemical reaction in skin cells that leads to increased production of the pigment melanin. This reaction can occur only if a specific molecule in the cell absorbs a certain minimum amount of energy. A short wavelength ultraviolet photon has enough energy to trigger the reaction, but a longer-wavelength visible-light photon does not. Hence ultraviolet light causes tanning, while visible light cannot.