Chapter 2: Particle Properties of Waves

• Particle: Mechanics

• Waves: Optics. Carry energy and momentum from one place to another.

• Microscopic world of atoms and molecules, electrons and nuclei: in this world, there are

neither particles nor waves in our sense.

• We regard electrons as particles because they possess charge and mass and behave according

to the laws of particle mechanics

◦ However, that it is just as correct to interpret a moving electron as a wave as it is to

interpret it as a particle.

• We regard electromagnetic waves as waves because under suitable circumstances they

exhibit diffraction, interference, and polarization.

◦ Similarly, we shall see that under other circumstances electromagnetic waves behave as

though they consist of streams of particles.

• The wave-particle duality is central to an understanding of modern physics.

2.1 ELECTROMAGNETIC WAVES

• Coupled electric and magnetic oscillations that move with the speed of light and exhibit

typical wave behavior.

• Accelerated electric charges generate linked electric and magnetic disturbances that can

travel indefinitely through space.

• If the charges oscillate periodically, the disturbances are waves whose electric and

magnetic components are perpendicular to each other and to the direction of propagation, as

in Fig. 2.1.

Figure 2.1 The electric and magnetic fields in an electromagnetic wave vary together. The fields are perpendicular

to each other and to the direction of propagation of the wave.

James Clerk

Maxwell

(1831–1879)

Maxwell was able to show that the speed c of electromagnetic waves in

free space is given by

where ε

the electric permittivity of free space and µ

is its magnetic

permeability. This is the same as the speed of light waves.

• Many features of EM waves interaction with matter depend upon their frequencies. Light

waves, which are em waves the eye responds to, span only a brief frequency interval, from

about 4.3x10

Hz for red light to about 7.5x10

Hz for violet light.

Figure 2.2 The spectrum of electromagnetic radiation.

• Principle of superposition. When two or more waves of the same nature travel past a point at

the same time, the instantaneous amplitude there is the sum of the instantaneous amplitudes

of the individual waves.

• Amplitude of a water wave is the height of the water surface relative to its normal level.

• Amplitude of a sound wave is the change in pressure relative to the normal pressure.

• Amplitude of a light wave (E=cB) can be taken as either E or B. Usually E is used, since its

interactions with matter give rise to nearly all common optical effects.

• When two or more trains of light waves meet in a region, they interfere to produce a new

wave there whose instantaneous amplitude is the sum of those of the original waves.

◦ Constructive interference refers to the reinforcement of waves with the same phase to

produce a greater amplitude. Fig 2.3a

◦ Destructive interference refers to the partial or complete cancellation of waves whose

phases differ. Fig 2.3b

◦ If the original waves have different frequencies, the result will be a mixture of

constructive and destructive interference.

Figure 2.3 (a) In constructive interference, superposed waves in phase reinforce each other. (b) In destructive

interference, waves out of phase partially or completely cancel each other.

• Young’s Double Slit Experiment. Interference of light waves Fig. 2.4.

• From each slit secondary waves spread out as though originating at the

slit. Diffraction.

• At those places on the screen where the path lengths from the two slits

differ by an odd number of half wavelengths, λ (1/2, 3/2, 5/2, …),

destructive interference occurs and a dark line is the result.

• At those places where the path lengths are equal or differ by a whole

number of wavelengths, λ (1, 2, 3, …), constructive interference occurs

and a bright line is the result.

• At intermediate places the interference is only partial, so the light intensity

on the screen varies gradually between the bright and dark lines.

Thomas Young

1773-1829

Figure 2.4 Origin of the interference pattern in Young’s experiment.

• Interference and diffraction are found only in waves.

• If light consisted of a stream of classical particles, the entire screen would be dark.

• Thus, Young’s experiment is proof that light consists of waves.

2.2 BLACKBODY RADIATION

• Only the quantum theory of light can explain its origin.

• Until the end of the nineteenth century the nature of light seemed settled forever.

• Attempts to understand the origin of the radiation emitted by bodies of matter.

• All objects radiate such energy continuously whatever their temperatures, though which

frequencies predominate depends on the temperature. At room temperature, most of the

radiation is in the infrared part of the spectrum and hence is invisible.

• The ability of a body to radiate is closely related to its ability to absorb radiation. This is to

be expected, since a body at a constant temperature is in thermal equilibrium with its

surroundings and must absorb energy from them at the same rate as it emits energy.

• Blackbody. Ideal body that absorbs all radiation incident upon it, regardless of frequency.

• In the laboratory, a blackbody can be approximated by a hollow object with a very small

hole leading to its interior (Fig. 2.5).

• Any radiation striking the hole enters the cavity,

where it is trapped by reflection back and forth until it

is absorbed.

• The cavity walls are constantly emitting and

absorbing radiation, and it is in the properties of this

radiation (blackbody radiation).

• Sample blackbody radiation simply by inspecting

what emerges from the hole in the cavity.

• A blackbody radiates more when it is hot than when it

is cold, and the spectrum of a hot blackbody has its

peak at a higher frequency than the peak in the

spectrum of a cooler one.

• The spectrum of blackbody radiation is shown in Fig.

2.6 for two temperatures.

Figure 2.5 A hole in the wall of a hollow object is an excellent approximation of a

blackbody.

Figure 2.6 Blackbody spectra. The spectral distribution of energy in the radiation

depends only on the temperature of the body. The higher the temperature, the greater

the amount of radiation and the higher the frequency at which the maximum emission

occurs.

The Ultraviolet Catastrophe

• Why does the blackbody spectrum have the shape shown in Fig.

2.6?

• Lord Rayleigh and James Jeans started by considering the

radiation inside a cavity of absolute temperature to be a series of

standing em waves (Fig. 2.7).

• The condition for standing waves in such a cavity is that the path

length from wall to wall, whatever the direction, must be a whole

number of half-wavelengths, so that a node occurs at each reflecting

surface.

• The number of independent standing waves in the frequency

interval per unit volume in the cavity turned out to be

Figure 2.7 Standing waves that have nodes at the walls, which restricts their possible

wavelengths. Shown are three possible wavelengths when the distance between opposite

walls is L.

• Because each standing wave in a cavity originates in an oscillating electric charge in the

cavity wall, two degrees of freedom are associated with the wave and it should have an

average energy of

• The total energy per unit volume in the cavity in the frequency interval is therefore

• The Rayleigh-Jeans formulae, contains everything that classical physics can say about the

spectrum of blackbody radiation.

• Eq. (2.3) shows that it cannot possibly be correct.

• In reality, of course, the energy density (and radiation rate) falls to 0 as frequency goes to

infinity (see Fig. 2.8). This discrepancy became known as the ultraviolet catastrophe of

classical physics.

• Where did Rayleigh and Jeans go wrong?

Figure 2.8 Comparison of the Rayleigh-Jeans

formula for the spectrum of the radiation from a

blackbody at 1500 K with the observed spectrum.

Max Planck

(1858–1947)

Nobel Prize in Physics in 1918

• This failure of classical physics led Planck to the discovery that radiation is emitted in

quanta whose energy is hν. Nobel Prize in 1918.

• Considered to mark the start of modern physics.

• At high frequencies and at low frequencies Planck’s formula becomes Rayleigh-Jeans

formula.

• The oscillators in the cavity walls could not have a continuous distribution of possible

energies ε but must have only the specific energies

• A

n oscillator emits radiation of frequency ν when it drops from one energy state to the next

lower one, and it jumps to the next higher state when it absorbs radiation of frequency ν.

• Each discrete bundle of energy hν is called a quantum (plural quanta) from the Latin for

“how much.”

• With oscillator energies limited to nhν, the average energy per oscillator in the cavity walls

Example 2.1

Assume that a certain 660-Hz tuning fork can be considered as a harmonic oscillator whose

vibrational energy is 0.04 J. Compare the energy quanta of this tuning fork with those of an atomic

oscillator that emits and absorbs orange light whose frequency is 5.00x10

Hz.

2.3 PHOTOELECTRIC EFFECT

• The energies of electrons liberated by light depend on the frequency of the light.

• Electrons emitted when the frequency of the light was sufficiently high. This phenomenon is

known as the photoelectric effect and the emitted electrons are called photoelectrons.

• Some of the photoelectrons that emerge from

this surface have enough energy to reach the

cathode despite its negative polarity, and they

constitute the measured current.

• When the voltage is increased to a certain

value V

, of the order of several volts, no

more photoelectrons arrive, as indicated by

the current dropping to zero. Maximum

photoelectron kinetic energy.

Figure 2.9 Experimental observation of the photoelectric effect.

Three experimental findings;

1. Within the limits of experimental accuracy (about 10

-9

s), there is no time interval between

the arrival of light at a metal surface and the emission of photoelectrons. However, because

the energy in an em wave is supposed to be spread across the wavefronts, a period of time

should elapse before an individual electron accumulates enough energy (several eV) to

leave the metal.

2. A bright light yields more photoelectrons than a dim one of

the same frequency, but the electron energies remain the same

(Fig. 2.10). The em theory of light, on the contrary, predicts

that the more intense the light, the greater the energies of the

electrons.

Figure 2.10 Photoelectron current is proportional to light intensity I for all retarding

voltages. The stopping potential V

, which corresponds to the maximum

photoelectron energy, is the same for all intensities of light of the same frequency.

3. The higher the frequency of the light, the more energy the photoelectrons have (Fig. 2.11).

◦ At frequencies below a certain critical frequency ν

, which is characteristic of each

particular metal, no electrons are emitted.

◦ Above ν

the photoelectrons range in energy from 0 to a maximum value that increases

linearly with increasing frequency (Fig.2.12). This observation, also, cannot be

explained by the em theory of light.

Figure 2.11 The stopping

potential V

, and hence the

maximum photoelectron

energy, depends on the

frequency of the light. When

the retarding potential is V=0,

the photoelectron current is

the same for light of a given

intensity regardless of its

frequency.

Figure 2.12 Maximum photoelectron kinetic energy

max

versus frequency of incident light for three

metal surfaces.

Quantum Theory of Light

• In 1905, Einstein realized that the photoelectric effect

could be understood if the energy in light is not spread out

over wavefronts but is concentrated in small packets, or

photons. Nobel Prize in 1921.

• Each photon of light of frequency ν has the energy hν, the

same as Planck’s quantum energy.

Albert Einstein 1879 – 1955

Nobel Prize in Physics in 1921

• Energy was not only given to em waves in separate quanta but was also carried by the waves

in separate quanta.

• The three experimental observations listed above follow directly from Einstein’s hypothesis.

1. Because em wave energy is concentrated in photons and not spread out, there should be

no delay in the emission of photoelectrons.

2. All photons of frequency have the same energy, so changing the intensity of a

monochromatic light beam will change the number of photoelectrons but not their

energies.

3. The higher the frequency ν, the greater the photon energy hν and so the more energy the

photoelectrons have.

• There must be a minimum energy φ for an electron to escape from a particular metal surface

or else electrons would pour out all the time. This energy is called the work function of the

metal, and is related to ν

by the formula

• The greater the work function of a metal, the more energy is needed for an electron to leave

its surface, and the higher the critical frequency for photoelectric emission to occur.

• where hν is the photon energy, KE

max

is the maximum photoelectron energy (which is

proportional to the stopping potential), and φ is the minimum energy needed for an electron

to leave the metal.

Figure 2.13 If the energy hν

(the work function of the

surface) is needed to remove an electron from a metal

surface, the maximum electron kinetic energy will be

hν-hν

when light of frequency is directed at the surface.

• In terms of electron volts, the formula E=hν for photon energy becomes

Example 2.2

Ultraviolet light of wavelength 350 nm and intensity 1.00 W/m

is directed at a potassium surface.

(a) Find the maximum KE of the photoelectrons.

(b) If 0.50 percent of the incident photons produce photoelectrons, how many are emitted per

second if the potassium surface has an area of 1.00 cm

2.4 WHAT IS LIGHT?

• Both wave and particle.

◦ The wave theory of light explains diffraction and interference, which the quantum theory

cannot account for. According to the wave theory, light waves leave a source with their

energy spread out continuously through the wave pattern.

◦ The quantum theory explains the photoelectric effect, which the wave theory cannot

account for. According to the quantum theory, light consists of individual photons, each

small enough to be absorbed by a single electron.

• In double-slit interference pattern on a screen.

◦ When it passes through the slits, light is behaving as a wave does. When it strikes the

screen, light is behaving as a particle does. Apparently light travels as a wave but

absorbs and gives off energy as a series of particles.

◦ In the wave model, the light intensity at a place on the screen depends on E

, the average

over a complete cycle of the square of the instantaneous magnitude E of the em wave’s

electric field.

◦ In the particle model, this intensity depends instead on Nhν, where N is the number of

photons per second per unit area that reach the same place on the screen.

• See Fig 2.4. Number of photons.

• Think of light as having a dual character. The wave theory and the quantum theory

complement each other. Either theory by itself is only part of the story and can explain only

certain effects.

2.5 X-RAYS

• They consist of high-energy photons.

• The photoelectric effect provides convincing evidence that photons of light can transfer

energy to electrons. Is the inverse process also possible?

• That is, can part or all of the kinetic energy of a moving electron be converted into a

photon?

• In 1895 Wilhelm Roentgen found that a highly penetrating radiation of

unknown nature is produced when fast electrons impinge on matter.

• The faster the original electrons, the more penetrating the resulting x-

rays, and the greater the number of electrons, the greater the intensity

of the x-ray beam.

• X-rays are em waves. Electromagnetic theory predicts that an

accelerated electric charge will radiate em waves, and a rapidly

moving electron suddenly brought to rest is certainly accelerated.

• Radiation produced under these circumstances is given the German

name bremsstrahlung (“braking radiation”).

• Electromagnetic radiation with wavelengths from about 0.01 to about

10 nm falls into the category of x-rays.

• In 1912 a method was devised for measuring the wavelengths of x-

rays. A diffraction experiment.

Wilhelm Konrad

Roentgen

(1845–1923)

First Nobel Prize

in Physics in 1902

• The spacing between adjacent lines on a diffraction grating must be of the same order of

magnitude as the wavelength of the light for satisfactory results.

• Max von Laue realized that the wavelengths suggested for x-rays were comparable to the

spacing between adjacent atoms in crystals.

• He therefore proposed that crystals be used to diffract x-rays, with their regular lattices

acting as a kind of three-dimensional grating. Figure 2.15 is a diagram of an x-ray tube.

• Figure 2.15 is a diagram of an x-ray tube.

Figure 2.15 An x-ray tube. The higher the accelerating

voltage V, the faster the electrons and the shorter the

wavelengths of the x-rays.

• As we know, classical electromagnetic

theory predicts bremsstrahlung when

electrons are accelerated, which

accounts in general for the x-rays

produced by an x-ray tube.

• Figures 2.16 and 2.17 show the x-ray

spectra that result when tungsten and

molybdenum targets are bombarded by electrons at several different accelerating potentials.

Figure 2.16 X-ray spectra of

tungsten at various accelerating

potentials.

Figure 2.17 X-ray spectra

of tungsten and molybdenum at

35 kV accelerating potential.

The curves exhibit two

features electromagnetic

theory cannot explain:

• In the case of

molybdenum,

intensity peaks occur that indicate the enhanced

production of x-rays at certain wavelengths. These peaks occur at specific wavelengths for

each target material and originate in rearrangements of the electron structures of the target

atoms after having been disturbed by the bombarding electrons.

◦ The presence of x-rays of specific wavelengths, a decidedly non-classical effect, in

addition to a continuous x-ray spectrum.

• The x-rays produced at a given accelerating potential V vary in wavelength, but none has a

wavelength shorter than a certain value λ

min

. Increasing V decreases λ

min

. Duane and Hunt

found experimentally that

◦ The second observation fits in with the quantum theory of radiation. Most of the

electrons that strike the target undergo numerous glancing collisions, with their energy

going simply into heat.

◦ A few electrons, though, lose most or all of their energy in single collisions with target

atoms. This is the energy that becomes x-rays.

◦ X-rays production, then, represents an inverse photoelectric effect. Instead of photon

energy being transformed into electron KE, electron KE is being transformed into

photon energy.

• Since work functions are only a few electron-volts whereas the accelerating potentials in x-

ray tubes are typically tens or hundreds of thousands of volts, we can ignore the work

function and interpret the short wavelength limit of Eq. (2.12) as corresponding to the case

where the entire kinetic energy KE=Ve of a bombarding electron is given up to a single

photon of energy hν

max

which is the Duane-Hunt formula of Eq. (2.12).

Example 2.3

Find the shortest wavelength present in the radiation from an x-ray machine whose accelerating

potential is 50,000 V.

2.6 X-RAY DIFFRACTION

• How x-ray wavelengths can be determined.

• A crystal consists of a regular array of atoms, each of which can scatter em waves. The

mechanism of scattering is straightforward.

◦ An atom in a constant electric field becomes polarized since its negatively charged

electrons and positively charged nucleus experience forces in opposite directions. So,

the result is a distorted charge distribution equivalent to an electric dipole.

◦ In the presence of the alternating electric field of an em wave of frequency ν, the

polarization changes back and forth with the same frequency ν.

◦ An oscillating electric dipole is thus created at the expense of some of the energy of the

incoming wave.

◦ The oscillating dipole in turn radiates em waves of frequency ν, and these secondary

waves go out in all directions except along the dipole axis. (see Fig. 2.18)

Figure 2.18 The scattering of electromagnetic

radiation by a group of atoms. Incident plane waves

are reemitted as spherical waves.

• A monochromatic beam of x-rays that falls

upon a crystal will be scattered in all

directions inside it.

• However, owing to the regular arrangement of

the atoms, in certain directions the scattered

waves will constructively interfere with one

another while in others they will destructively

interfere.

• The atoms in a crystal may be thought of as defining families of parallel planes with each

family having a characteristic separation between its component planes.

• This analysis was suggested in 1913 by W. L Bragg, in honor of whom

the above planes are called Bragg planes.

• The conditions that

must be fulfilled for

radiation scattered by

crystal atoms to undergo

constructive interference

may be obtained from a

diagram like that in Fig.

2.20.

Figure 2.20 X-ray scattering from a cubic crystal.

William

Lawrence Bragg

(1890–1971)

Nobel Prize in

Physics in 1915

• A beam containing x-rays of wavelength λ is incident upon a crystal at an angle  with a

family of Bragg planes whose spacing is d.

• The beam goes past atom A in the first plane and atom B in the next, and each of them

scatters part of the beam in random directions.

• Constructive interference takes place only between those scattered rays that are parallel and

whose paths differ by exactly λ, 2λ, 3λ, and so on.

• That is, the path difference must be nλ, where n is an integer.

• Conditions;

1. The first condition on I and II is that their common scattering angle be equal to the angle

of incidence  of the original beam.

2. The second condition is that

since ray II must travel the distance 2d sin farther than ray I. The integer n is the order

of the scattered beam.

Figure 2.21 X-ray spectrometer.

• If the spacing d between adjacent

Bragg planes in the crystal is known,

the x-ray wavelength λ may be

calculated.

2.7 COMPTON EFFECT

• Further confirmation of the photon model.

• According to the quantum theory of light, photons behave like particles except for their lack

of rest mass.

• Figure 2.22 shows a collision: an x-ray photon strikes an electron (assumed to be initially at

rest in the laboratory coordinate system).

• We can think of the photon as losing an amount of energy in the collision that is the same as

the kinetic energy KE gained by the electron.

Figure 2.22 (a) The scattering of a photon by an electron is called the Compton effect. Energy and

momentum are conserved in such an event, and as a result the scattered photon has less energy

(longer wavelength) than the incident photon. (b) Vector diagram of the momenta and their

components of the incident and scattered photons and the scattered electron.

the momentum of a massless particle is related to its energy by the formula

Since the energy of a photon is hν, its momentum is

• In the collision momentum must be conserved in each of two mutually perpendicular

directions.

◦

In the original photon direction

◦ and in perpendicular to original photon direction

• The angle  is that between the directions of the initial and scattered photons, and  is that

between the directions of the initial photon and the recoil electron.

• From Eqs. (2.14), (2.16), and (2.17) we can find a formula that relates the wavelength

difference between initial and scattered photons with the angle  between their directions,

both of which are readily measurable quantities (unlike the energy and momentum of the

recoil electron).

• Equation (2.23) was derived by Arthur H. Compton in the early 1920s, and

the phenomenon it describes, which he was the first to observe, is known as

the Compton effect.

• It constitutes very strong evidence in support of the quantum theory of

radiation.

• Change in wavelength is independent of the wavelength  of the incident

photon.

• The quantity

is called the Compton wavelength of the scattering particle. For an electron

=2.426x10

-12

m, which is 2.426 pm (1 pm=1 picometer=10

-12

m).

Arthur Holly

Compton (1892–

1962) Nobel Prize

in Physics in 1927

• The Compton wavelength gives the scale of the wavelength change of the incident photon.

From (2.23) we note that the greatest wavelength change possible corresponds to =180°.

• Changes of this magnitude or less are readily observable only in x-rays.

• The Compton effect is the chief means by which x-rays lose energy when they pass through

matter.

• The experimental demonstration of the Compton effect is straightforward. As in Fig. 2.23, a

beam of x-rays of a single, known wavelength is

directed at a target, and the wavelengths of the

scattered x-rays are determined at various angles

• The results are shown in Fig. 2.24.

Figure 2.23 Experimental demonstration of the Compton effect.

Figure 2.24 Experimental confirmation of Compton scattering. The greater the scattering angle, the greater the

wavelength change, in accord with Eq. (2.23).

Example 2.4

X-rays of wavelength 10.0 pm are scattered from a target. (a) Find the wavelength of the x-rays

scattered through 45°. (b) Find the maximum wavelength present in the scattered x-rays. (c)

Find the maximum kinetic energy of the recoil electrons.

2.8 PAIR PRODUCTION

1. Energy into matter.

2. As we have seen, in a collision a photon can give an electron all of its energy (the

photoelectric effect) or only part (the Compton effect).

3. It is also possible for a photon to materialize into an electron and a positron, which is a

positively charged electron. In this process, called pair production, electromagnetic energy is

converted into matter.

Figure 2.25 In the process of

pair production, a photon of sufficient energy materializes into an electron and a positron.

• No conservation principles are violated when an electron-positron pair is created near an

atomic nucleus (Fig. 2.25).

◦ The sum of the charges of the electron (q=-e) and of the positron (q=e) is zero, as is the

charge of the photon

◦ the total energy, including rest energy, of the electron and positron equals the photon

energy

◦ linear momentum is conserved with the help of the nucleus, which carries away enough

photon momentum for the process to occur.

• The rest energy mc

of an electron or positron is 0.51 MeV, hence pair production requires a

photon energy of at least 1.02 MeV.

• Any additional photon energy becomes kinetic energy of the electron and positron. The

corresponding maximum photon wavelength is 1.2 pm. Electromagnetic waves with such

wavelengths are called gamma rays, symbol γ, and are found in nature as one of the

emissions from radioactive nuclei and in cosmic rays.

• The inverse of pair production occurs when a positron is near an electron and the two come

together under the influence of their opposite electric charges. Both particles vanish

simultaneously, with the lost mass becoming energy in the form of two gamma-ray photons:

• The total mass of the positron and electron is equivalent to 1.02 MeV, and each photon has

an energy hν of 0.51 MeV plus half the kinetic energy of the particles relative to their center

of mass.

• The directions of the photons are such as to conserve both energy and linear momentum, and

no nucleus or other particle is needed for this pair annihilation to take place.

Photon Absorption

• The three chief ways in which photons of light, x-rays, and gamma rays interact with matter

are summarized in Fig. 2.27.

• In all cases photon energy is transferred

to electrons which in turn lose energy to

atoms in the absorbing material.

• At low photon energies, the

photoelectric effect is the main

mechanism of energy loss.

• The importance of the photoelectric

effect decreases with increasing energy,

to be succeeded by Compton scattering.

Figure 2.27 X- and gamma rays interact with matter chiefly through the photoelectric effect, Compton scattering,

and pair production. Pair production requires a photon energy of at least 1.02 MeV.

• The greater the atomic number of the absorber,

the higher the energy at which the photoelectric

effect remains significant.

• In the lighter elements, Compton scattering

becomes dominant at photon energies of a few

tens of keV,

• whereas in the heavier ones this does not happen

until photon energies of nearly 1 MeV are

reached (Fig. 2.28).

Figure 2.28 The relative probabilities of the photoelectric effect,

Compton scattering, and pair production as functions of energy in

carbon (a light element) and lead (a heavy element).