# And the Photoelectric

Effect a

The focus in this book is on the interaction of electromagnetic waves with electrons contained in the pigments, binders, and support that comprise a work of art.1 These waves have wavelengths that range from 1 nanometer (10~9 meter) for x-rays to 1 kilometer (103 meters) for radio waves. Wavelengths of visible light are in the range of 400 to 700 nanometers (nm). In this appendix we consider the formulation of photon wavelength and energy as well as the absorption of a photon by an electron.

Moving waves can be described by their frequency, or number of waves per second, as well as by their wavelength. If a train of waves is moving past a stationary observer at a frequency of f complete cycles, or waves, per second, and if each of these has a wavelength of A centimeters per cycle, then the wave must be moving at a speed of v cm per second. The velocity (v), wavelength (A), and frequency (f) are related by velocity = frequency | ——| X wavelength(distance) = distance, time time v :

All electromagnetic radiation moves in a vacuum at a universal speed. This is the speed of light, c = 30,000,000,000 centimeters per second (usually written in powers of ten, c = 3 X 1010 cm/sec). The

'Information is also available on the Internet in Readings and Activities in Patterns in Nature. http://acept.la.asu.edu./PiN

universal speed of light in a vacuum goes against our intuition: We would expect that high-energy (short wavelength) radiation would move faster than low-energy (long wavelength) radiation. We can consider light as a stream of minute packets of energy, photons, which create a pulsating electromagnetic disturbance. A single photon differs from another photon only by its energy. In empty space (vacuum) all photons travel with the same velocity. Photons are slowed down when they pass through different media such as water, glass, or even air. This slowing down accounts for the refraction, or bending, of light by optical lenses. The energy of the photon is not changed, but the wavelength is. Photons of different energies are slowed by different amounts, which leads to the dispersion of light and the appearance of rainbows.

Because the speed of light in a vacuum is constant, if we know either the frequency or the wavelength of electromagnetic radiation we can calculate the other quantity from equation A.1. The frequency of the electromagnetic vibration we call light is given in units of "hertz" (abbreviated Hz and named after Heinrich Hertz (1857-1894) a famous German investigator of electromagnetism). One hertz is one vibration per second; the range of the pure spectrum perceived by the eye extends from about 4.3 X 1012 Hz in the red range to about 7.5 X 1012 Hz in the violet.

This aspect of the discussion suggests that light, and other electromagnetic radiation, is composed of waves. It was very disturbing, therefore, when phenomena were discovered (around 1900) that clearly indicated that light is made up of particles, called photons. One such phenomenon involved the photoelectric effect. It was known that if one shines a beam of light on a clean surface of a metal, electrons will be ejected from the metal. The process is shown in Figure A.1. The light has to exceed a certain energy to remove electrons from the metal surface. If the light has more than the minimum energy required, then the extra energy will be given to the ejected electrons as kinetic energy of motion.

In this discovery light of a single wavelength behaved as if it consisted of separate particles, photons, all with the same energy, with each ejected electron being the result of a collision between one photon and one electron in the metal. Greater intensity of light meant only that more photons were hitting the metal

Fig. A.1. In the photoelectric effect a photon incident on a surface (here a metal surface in a vacuum) transfers its energy to an electron, which leaves the surface and is detected. The photoelectric effect demonstrates the particle nature of light.

Appendix A Photons, Electrons, and Photoelectric Effect

per second and more electrons were ejected, not that there was more energy per photon. The energy of the outgoing electrons depended on the frequency of light used.

The energy (E) of the incoming photons was found to be directly proportional to the light frequency (f), which can be written as

in which h is a constant.

Max Planck first proposed this relationship between energy and frequency in 1900 as part of his study of the way in which heated solids emit radiation. The constant h is called Planck's constant in his honor and has a value h = 4.136 x 10~15 eV sec.

For photons, the property most readily measured is their energy. The different colors of light, for example, are thought of as representing photons of different energies. The link between the particle theory and the wave theory lies in Planck's fundamental postulate of quantum theory given by equation (A.2).

There is an inverse relation between the energy E of the photon and the wavelength:

The energy E in equation A.3 can be expressed in many units. In the analysis of art and in the description of light, the most convenient unit of energy to use is the electron volt, abbreviated eV. In terms of wavelength A in nanometers (nm) and energy E in electron volts (eV), equation (A.3) can be expressed for light traveling in a vacuum as

and shown in Figure A.2.

The constants and numerical relations in equations (A.3) and (A.4) are found from Planck's equation (A.2) by writing the relationship for frequency (f) in (A.1) as f =A, (A-5)

where c is the velocity of light in vacuum so that Planck's equation becomes

The value of Planck's constant h is 4.136 x 10-15 eV sec and the velocity of light c is 3 x 108m/sec, or 3 x 1017 nm/sec, so that hc 12.4 x 102, or 1240 eV nm.

We use nanometers (nm) and electron volts (eV) in this text. Other units for the wavelength A, the distance between two crests of the light wave, are centimeters (cm), micrometers or microns (um), where 1 micron = 10~4cm, and angstrom units (A), where 1 A = 10~8 cm.

Visible light is composed of photons in the energy range of around 2 to 3 eV (Chapter 3). Orange light with a wavelength of 620 nanometers is composed of photons with energy of 2 eV. It is the energy range of 2 to 3 eV that trig-ers the photoreceptors in the eye. Lower energies (longer wavelengths) are not detected by the human eye but can be detected by special semiconductor infrared sensors. Higher energies (shorter wavelengths) such as x-rays damage human tissue and are detected by x-ray sensitive photographic film or again by special semiconductor devices.

Light rays are composed of photons whose energy or wavelength specifies a color from red to violet. An increase in the intensity or brightness of a light source increases the number of photons per unit area, or flux. Brightness of light refers to the flux of photons not their energy.

Fig. A.2. A curve that shows the relationship E (eV) 1240/A (nm) given in (A.4) for the energy in eV versus photon wavelength in nanometers for the visible region of the elec tromagnetic spectrum.

Refraction, Reflection, and Dispersion

The velocity of light in vacuum is constant for all wavelengths of light. When light enters a medium such as glass, the velocity of light decreases. In glass there is a decrease in velocity of 33% (velocity changes from 3 to 2 X 1010cm/sec), and in other materials the decrease can be even more substantial, 50 to 60%.

A beam of light in air will change direction, be refracted, when entering glass at an angle to the surface normal; this bending of the beam is called refraction.1 Refraction is an effect that occurs when a light wave passes at an angle to a boundary from one medium to another in which there is a change in velocity. Consider the case where light enters a medium where the wave velocity decreases. The wave inside the new medium is moving slower; therefore, since it remains connected to the faster-moving wave in the old medium, the wave turns, becoming more nearly parallel with the boundary, as shown in Figure B.1. The greater the change in velocity, the greater the change in direction.