Figure 29.7The photoelectric effect can be observed by allowing light to fall on the metal plate in this evacuated tube. Electrons ejected by the light are collected on the
collector wire and measured as a current. A retarding voltage between the collector wire and plate can then be adjusted so as to determine the energy of the ejected electrons.
For example, if it is sufficiently negative, no electrons will reach the wire. (credit: P.P. Urone)
This effect has been known for more than a century and can be studied using a device such as that shown inFigure 29.7. This figure shows an
evacuated tube with a metal plate and a collector wire that are connected by a variable voltage source, with the collector more negative than the
plate. When light (or other EM radiation) strikes the plate in the evacuated tube, it may eject electrons. If the electrons have energy in electron volts
(eV) greater than the potential difference between the plate and the wire in volts, some electrons will be collected on the wire. Since the electron
energy in eV is
is the electron charge and
is the potential difference, the electron energy can be measured by adjusting the
retarding voltage between the wire and the plate. The voltage that stops the electrons from reaching the wire equals the energy in eV. For example, if
barely stops the electrons, their energy is 3.00 eV. The number of electrons ejected can be determined by measuring the current between
the wire and plate. The more light, the more electrons; a little circuitry allows this device to be used as a light meter.
What is really important about the photoelectric effect is what Albert Einstein deduced from it. Einstein realized that there were several characteristics
of the photoelectric effect that could be explained only ifEM radiation is itself quantized: the apparently continuous stream of energy in an EM wave is
actually composed of energy quanta called photons. In his explanation of the photoelectric effect, Einstein defined a quantized unit or quantum of EM
energy, which we now call aphoton, with an energy proportional to the frequency of EM radiation. In equation form, thephoton energyis
is the energy of a photon of frequency
is Planck’s constant. This revolutionary idea looks similar to Planck’s quantization of
energy states in blackbody oscillators, but it is quite different. It is the quantization of EM radiation itself. EM waves are composed of photons and are
not continuous smooth waves as described in previous chapters on optics. Their energy is absorbed and emitted in lumps, not continuously. This is
exactly consistent with Planck’s quantization of energy levels in blackbody oscillators, since these oscillators increase and decrease their energy in
by absorbing and emitting photons having
. We do not observe this with our eyes, because there are so many photons in
common light sources that individual photons go unnoticed. (SeeFigure 29.8.) The next section of the text (Photon Energies and the
Electromagnetic Spectrum) is devoted to a discussion of photons and some of their characteristics and implications. For now, we will use the
photon concept to explain the photoelectric effect, much as Einstein did.
Figure 29.8An EM wave of frequency
is composed of photons, or individual quanta of EM radiation. The energy of each photon is
is the frequency of the EM radiation. Higher intensity means more photons per unit area. The flashlight emits large numbers of photons of many different
frequencies, hence others have energy
, and so on.
The photoelectric effect has the properties discussed below. All these properties are consistent with the idea that individual photons of EM radiation
are absorbed by individual electrons in a material, with the electron gaining the photon’s energy. Some of these properties are inconsistent with the
idea that EM radiation is a simple wave. For simplicity, let us consider what happens with monochromatic EM radiation in which all photons have the
1. If we vary the frequency of the EM radiation falling on a material, we find the following: For a given material, there is a threshold frequency
for the EM radiation below which no electrons are ejected, regardless of intensity. Individual photons interact with individual electrons. Thus if
the photon energy is too small to break an electron away, no electrons will be ejected. If EM radiation was a simple wave, sufficient energy
could be obtained by increasing the intensity.
2. Once EM radiation falls on a material, electrons are ejected without delay. As soon as an individual photon of a sufficiently high frequency is
absorbed by an individual electron, the electron is ejected. If the EM radiation were a simple wave, several minutes would be required for
sufficient energy to be deposited to the metal surface to eject an electron.
3. The number of electrons ejected per unit time is proportional to the intensity of the EM radiation and to no other characteristic. High-intensity EM
radiation consists of large numbers of photons per unit area, with all photons having the same characteristic energy
4. If we vary the intensity of the EM radiation and measure the energy of ejected electrons, we find the following:The maximum kinetic energy of
ejected electrons is independent of the intensity of the EM radiation. Since there are so many electrons in a material, it is extremely unlikely that
two photons will interact with the same electron at the same time, thereby increasing the energy given it. Instead (as noted in 3 above),
increased intensity results in more electrons of the same energy being ejected. If EM radiation were a simple wave, a higher intensity could give
more energy, and higher-energy electrons would be ejected.
CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS S 1033
5. The kinetic energy of an ejected electron equals the photon energy minus the binding energy of the electron in the specific material. An
individual photon can give all of its energy to an electron. The photon’s energy is partly used to break the electron away from the material. The
remainder goes into the ejected electron’s kinetic energy. In equation form, this is given by
is the maximum kinetic energy of the ejected electron,
is the photon’s energy, and BE is thebinding energyof the electron to
the particular material. (BE is sometimes called thework functionof the material.) This equation, due to Einstein in 1905, explains the properties
of the photoelectric effect quantitatively. An individual photon of EM radiation (it does not come any other way) interacts with an individual
electron, supplying enough energy, BE, to break it away, with the remainder going to kinetic energy. The binding energy is
is the threshold frequency for the particular material.Figure 29.9shows a graph of maximum
versus the frequency of incident EM
radiation falling on a particular material.
Figure 29.9Photoelectric effect. A graph of the kinetic energy of an ejected electron,
e, versus the frequency of EM radiation impinging on a certain material. There is a
threshold frequency below which no electrons are ejected, because the individual photon interacting with an individual electron has insufficient energy to break it away. Above
the threshold energy,
increases linearly with
, consistent with
. The slope of this line is
—the data can be used to determine Planck’s
constant experimentally. Einstein gave the first successful explanation of such data by proposing the idea of photons—quanta of EM radiation.
Einstein’s idea that EM radiation is quantized was crucial to the beginnings of quantum mechanics. It is a far more general concept than its
explanation of the photoelectric effect might imply. All EM radiation can also be modeled in the form of photons, and the characteristics of EM
radiation are entirely consistent with this fact. (As we will see in the next section, many aspects of EM radiation, such as the hazards of ultraviolet
(UV) radiation, can be explainedonlyby photon properties.) More famous for modern relativity, Einstein planted an important seed for quantum
mechanics in 1905, the same year he published his first paper on special relativity. His explanation of the photoelectric effect was the basis for the
Nobel Prize awarded to him in 1921. Although his other contributions to theoretical physics were also noted in that award, special and general
relativity were not fully recognized in spite of having been partially verified by experiment by 1921. Although hero-worshipped, this great man never
received Nobel recognition for his most famous work—relativity.
Example 29.1Calculating Photon Energy and the Photoelectric Effect: A Violet Light
(a) What is the energy in joules and electron volts of a photon of 420-nm violet light? (b) What is the maximum kinetic energy of electrons ejected
from calcium by 420-nm violet light, given that the binding energy (or work function) of electrons for calcium metal is 2.71 eV?
To solve part (a), note that the energy of a photon is given by
. For part (b), once the energy of the photon is calculated, it is a
straightforward application of
to find the ejected electron’s maximum kinetic energy, since BE is given.
Solution for (a)
Photon energy is given by
Since we are given the wavelength rather than the frequency, we solve the familiar relationship
for the frequency, yielding
Combining these two equations gives the useful relationship
Now substituting known values yields
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Converting to eV, the energy of the photon is
Solution for (b)
Finding the kinetic energy of the ejected electron is now a simple application of the equation
. Substituting the photon energy
and binding energy yields
=hf– BE=2.96 eV – 2.71 eV=0.246 eV.
The energy of this 420-nm photon of violet light is a tiny fraction of a joule, and so it is no wonder that a single photon would be difficult for us to
sense directly—humans are more attuned to energies on the order of joules. But looking at the energy in electron volts, we can see that this
photon has enough energy to affect atoms and molecules. A DNA molecule can be broken with about 1 eV of energy, for example, and typical
atomic and molecular energies are on the order of eV, so that the UV photon in this example could have biological effects. The ejected electron
(called aphotoelectron) has a rather low energy, and it would not travel far, except in a vacuum. The electron would be stopped by a retarding
potential of but 0.26 eV. In fact, if the photon wavelength were longer and its energy less than 2.71 eV, then the formula would give a negative
kinetic energy, an impossibility. This simply means that the 420-nm photons with their 2.96-eV energy are not much above the frequency
threshold. You can show for yourself that the threshold wavelength is 459 nm (blue light). This means that if calcium metal is used in a light
meter, the meter will be insensitive to wavelengths longer than those of blue light. Such a light meter would be completely insensitive to red light,
PhET Explorations: Photoelectric Effect
See how light knocks electrons off a metal target, and recreate the experiment that spawned the field of quantum mechanics.
Figure 29.10Photoelectric Effect (http://cnx.org/content/m42558/1.4/photoelectric_en.jar)
29.3Photon Energies and the Electromagnetic Spectrum
A photon is a quantum of EM radiation. Its energy is given by
and is related to the frequency
of the radiation by
(energy of a photon),
is the energy of a single photon and
is the speed of light. When working with small systems, energy in eV is often useful. Note that
Planck’s constant in these units is
Since many wavelengths are stated in nanometers (nm), it is also useful to know that
These will make many calculations a little easier.
All EM radiation is composed of photons.Figure 29.11shows various divisions of the EM spectrum plotted against wavelength, frequency, and
photon energy. Previously in this book, photon characteristics were alluded to in the discussion of some of the characteristics of UV, x rays, and
rays, the first of which start with frequencies just above violet in the visible spectrum. It was noted that these types of EM radiation have
characteristics much different than visible light. We can now see that such properties arise because photon energy is larger at high frequencies.
CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS S 1035
Figure 29.11The EM spectrum, showing major categories as a function of photon energy in eV, as well as wavelength and frequency. Certain characteristics of EM radiation
are directly attributable to photon energy alone.
Table 29.1Representative Energies for Submicroscopic
Effects (Order of Magnitude Only)
Rotational energies of molecules
Vibrational energies of molecules
Energy between outer electron shells in atoms
Binding energy of a weakly bound molecule
Energy of red light
Binding energy of a tightly bound molecule
Energy to ionize atom or molecule
10 to 1000 eV
Photons act as individual quanta and interact with individual electrons, atoms, molecules, and so on. The energy a photon carries is, thus, crucial to
the effects it has.Table 29.1lists representative submicroscopic energies in eV. When we compare photon energies from the EM spectrum inFigure
29.11with energies in the table, we can see how effects vary with the type of EM radiation.
Gamma rays, a form of nuclear and cosmic EM radiation, can have the highest frequencies and, hence, the highest photon energies in the EM
spectrum. For example, a
-ray photon with
has an energy
This is sufficient energy to
ionize thousands of atoms and molecules, since only 10 to 1000 eV are needed per ionization. In fact,
rays are one type ofionizing radiation, as
are x rays and UV, because they produce ionization in materials that absorb them. Because so much ionization can be produced, a single
photon can cause significant damage to biological tissue, killing cells or damaging their ability to properly reproduce. When cell reproduction is
disrupted, the result can be cancer, one of the known effects of exposure to ionizing radiation. Since cancer cells are rapidly reproducing, they are
exceptionally sensitive to the disruption produced by ionizing radiation. This means that ionizing radiation has positive uses in cancer treatment as
well as risks in producing cancer.
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Figure 29.12One of the first x-ray images, taken by Röentgen himself. The hand belongs to Bertha Röentgen, his wife. (credit: Wilhelm Conrad Röntgen, via Wikimedia
High photon energy also enables
rays to penetrate materials, since a collision with a single atom or molecule is unlikely to absorb all the
energy. This can make
rays useful as a probe, and they are sometimes used in medical imaging.x rays, as you can see inFigure 29.11, overlap
with the low-frequency end of the
ray range. Since x rays have energies of keV and up, individual x-ray photons also can produce large amounts
of ionization. At lower photon energies, x rays are not as penetrating as
rays and are slightly less hazardous. X rays are ideal for medical imaging,
their most common use, and a fact that was recognized immediately upon their discovery in 1895 by the German physicist W. C. Roentgen
(1845–1923). (SeeFigure 29.12.) Within one year of their discovery, x rays (for a time called Roentgen rays) were used for medical diagnostics.
Roentgen received the 1901 Nobel Prize for the discovery of x rays.
Connections: Conservation of Energy
Once again, we find that conservation of energy allows us to consider the initial and final forms that energy takes, without having to make
detailed calculations of the intermediate steps.Example 29.2is solved by considering only the initial and final forms of energy.
Figure 29.13X rays are produced when energetic electrons strike the copper anode of this cathode ray tube (CRT). Electrons (shown here as separate particles) interact
individually with the material they strike, sometimes producing photons of EM radiation.
CHAPTER 29 | INTRODUCTION TO QUANTUM PHYSICS S 1037
rays originate in nuclear decay, x rays are produced by the process shown inFigure 29.13. Electrons ejected by thermal agitation from a hot
filament in a vacuum tube are accelerated through a high voltage, gaining kinetic energy from the electrical potential energy. When they strike the
anode, the electrons convert their kinetic energy to a variety of forms, including thermal energy. But since an accelerated charge radiates EM waves,
and since the electrons act individually, photons are also produced. Some of these x-ray photons obtain the kinetic energy of the electron. The
accelerated electrons originate at the cathode, so such a tube is called a cathode ray tube (CRT), and various versions of them are found in older TV
and computer screens as well as in x-ray machines.
Example 29.2X-ray Photon Energy and X-ray Tube Voltage
Find the maximum energy in eV of an x-ray photon produced by electrons accelerated through a potential difference of 50.0 kV in a CRT like the
one inFigure 29.13.
Electrons can give all of their kinetic energy to a single photon when they strike the anode of a CRT. (This is something like the photoelectric
effect in reverse.) The kinetic energy of the electron comes from electrical potential energy. Thus we can simply equate the maximum photon
energy to the electrical potential energy—that is,
(We do not have to calculate each step from beginning to end if we know that all of
the starting energy
is converted to the final form
The maximum photon energy is
is the charge of the electron and
is the accelerating voltage. Thus,
From the definition of the electron volt, we know
1 J=1 C⋅V.
Gathering factors and converting energy to eV
)(1 eV)=50.0 keV.
This example produces a result that can be applied to many similar situations. If you accelerate a single elementary charge, like that of an
electron, through a potential given in volts, then its energy in eV has the same numerical value. Thus a 50.0-kV potential generates 50.0 keV
electrons, which in turn can produce photons with a maximum energy of 50 keV. Similarly, a 100-kV potential in an x-ray tube can generate up to
100-keV x-ray photons. Many x-ray tubes have adjustable voltages so that various energy x rays with differing energies, and therefore differing
abilities to penetrate, can be generated.
Figure 29.14X-ray spectrum obtained when energetic electrons strike a material. The smooth part of the spectrum is bremsstrahlung, while the peaks are characteristic of the
anode material. Both are atomic processes that produce energetic photons known as x-ray photons.
Figure 29.14shows the spectrum of x rays obtained from an x-ray tube. There are two distinct features to the spectrum. First, the smooth distribution
results from electrons being decelerated in the anode material. A curve like this is obtained by detecting many photons, and it is apparent that the
maximum energy is unlikely. This decelerating process produces radiation that is calledbremsstrahlung(German forbraking radiation). The second
feature is the existence of sharp peaks in the spectrum; these are calledcharacteristic x rays, since they are characteristic of the anode material.
Characteristic x rays come from atomic excitations unique to a given type of anode material. They are akin to lines in atomic spectra, implying the
energy levels of atoms are quantized. Phenomena such as discrete atomic spectra and characteristic x rays are explored further inAtomic Physics.
Ultraviolet radiation(approximately 4 eV to 300 eV) overlaps with the low end of the energy range of x rays, but UV is typically lower in energy. UV
comes from the de-excitation of atoms that may be part of a hot solid or gas. These atoms can be given energy that they later release as UV by
numerous processes, including electric discharge, nuclear explosion, thermal agitation, and exposure to x rays. A UV photon has sufficient energy to
ionize atoms and molecules, which makes its effects different from those of visible light. UV thus has some of the same biological effects as
and x rays. For example, it can cause skin cancer and is used as a sterilizer. The major difference is that several UV photons are required to disrupt
cell reproduction or kill a bacterium, whereas single
-ray and X-ray photons can do the same damage. But since UV does have the energy to alter
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