Chapter 2

Particle Properties of Waves

2 Particle Properties of Waves

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2 Particle Properties of Waves

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•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, 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

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•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 (EM)

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.

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2.1 Electromagnetic 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,

Figure 2.2 The spectrum of electromagnetic radiation.

•from about 4.3x10

Hz for red

light

•to about 7.5x10

Hz for violet

light.

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2.1 Electromagnetic Waves: 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.

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2.1 Electromagnetic Waves: Interference

• 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.

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2.1 Electromagnetic Waves: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.

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2.1 Electromagnetic Waves

•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.

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2.2 Blackbody Radiation

•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 energy continuously whatever their temperatures.

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.

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

an excellent approximation of a blackbody.

•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.

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2.2 Blackbody Radiation

•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.

•Why does the blackbody spectrum have the

shape shown in Fig. 2.6?

•Only the quantum theory of light can explain

its origin.

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 an”d the higher the frequency at which

the maximum emission occurs.

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2.2 Blackbody Radiation: The Ultraviolet Catastrophe

•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

•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

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

(Density of standing waves in cavity)

(Classical average energy per standing wave)

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2.2 Blackbody Radiation: The Ultraviolet Catastrophe

• 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.

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.

•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?

(Rayleigh-Jeans formula)

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2.2 Blackbody Radiation: The Ultraviolet Catastrophe

• This failure of classical physics led Planck to the discovery

that radiation is emitted in quanta whose energy is hν.

•Considered to mark the start of modern physics.

•At high frequencies and at low frequencies Planck’s formula

becomes Rayleigh-Jeans formula.

(Planck’s constant)

(Planck radiation formula)

Max Planck

(1858–1947)

Nobel Prize in

Physics in 1918

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

distribution of possible energies ε but must have only the specific

energies

•An 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.”

(Oscillator energies)

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2.2 Blackbody Radiation: The Ultraviolet Catastrophe

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

in the cavity walls

(Actual average energy per standing wave)

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.

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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 (the measured current).

•When the voltage is increased to a certain

value V

, no more photoelectrons arrive (the

current dropping to zero).

•Maximum photoelectron kinetic energy.

Figure 2.9 Experimental observation of the photoelectric

effect..

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2.3 Photoelectric Effect

•Three experimental findings;

1.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.

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.

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.

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2.3 Photoelectric Effect

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 2.12 Maximum photoelectron kinetic energy KE

max

versus

frequency of incident light for three

metal surfaces.

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.

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2.3 Photoelectric Effect; 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.

•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.

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2.3 Photoelectric Effect; Quantum Theory of Light

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

particular metal surface (Fig 2.13).

•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.

(Photoelectric effect)

(Work function)

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

(Photon energy)

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2.3 Photoelectric Effect; Quantum Theory of Light

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

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2.4 What is Light?

•Both wave and particle.

•The wave theory of light explains diffraction and interference, for

which the quantum theory cannot account.

•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, for which the

wave theory cannot account.

•According to the quantum theory, light consists of individual

photons, each small enough to be absorbed by a single electron.

•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.

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2.4 What is Light?

•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 on Nhν,

where N is the number of photons per second per unit

area that reach the same place on the screen.

Figure 2.14 (a) The wave

theory of light explains

diffraction and interference,

which the quantum theory

cannot account for.

(b) The quantum theory

explains the photoelectric

effect, which the wave

theory cannot account for.

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2.4 X-Rays: X-rays are EM waves

•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.

•The greater the number of electrons, the greater the intensity

of the x-ray beam.

•Electromagnetic theory predicts that an accelerated electric

charge will radiate EM waves.

• Acceleration: Rapidly moving electron suddenly

brought to rest.

Wilhelm Konrad

Roentgen

(1845–1923)

First Nobel Prize

in Physics in 1902

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2.4 X-Rays: Bremsstrahlung (“braking radiation”)

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

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

the same order of magnitude as the wavelength of the light.

•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 (Fig. 2.15).

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.

•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.

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2.4 X-Rays

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:

1.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.

2.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

(X-ray production)

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2.4 X-Rays: Inverse photoelectric effect

•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 lose most or all of their energy in single collisions with

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

•Since work functions are only a few electron-volts (eVs) whereas the

accelerating potentials in x-ray tubes are typically tens or hundreds of

thousands of volts, we can ignore the work function.

•Iinterpret the short wavelength limit of X-ray production Equation 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

(Duane-Hunt formula of X-ray production Equation)

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2.4 X-Rays

Example 2.3

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

accelerating potential is 50.000 V.

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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).

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2.6 X-Ray Diffraction: Bragg planes

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 .

•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.

William

Lawrence Bragg

(1890–1971)

Nobel Prize in

Physics in 1915

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2.6 X-Ray Diffraction: Conditions

Figure 2.20 X-ray scattering from a cubic

crystal.

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.

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.

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.

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2.6 X-Ray Diffraction: Experimental Demonstration

Figure 2.21 X-ray 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

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.

•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).

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

formula

2.7 Compton Effect: Conservations

•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.

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

(Photon momentum)

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

perpendicular directions.

(In the

original

photon

direction) (2)

(In

perpendicular

to original

photon

direction) (3)

•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.

(1)

From Eqns. (1-3) we can find a formula that relates ….

2.7 Compton Effect: Compton wavelength

•From Eqns. (1-3) 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).

(Compton effect)

•Compton effect equation was derived by Arthur H. Compton

in the early 1920s, and the phenomenon it describes 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

Arthur Holly

Compton

(1892–1962)

Nobel Prize in

Physics in 1927

(Compton wavelength)

is called the Compton wavelength of the scattering particle.

For an electron λ

=2.426x10

-12

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

-12

2.7 Compton Effect: Experimental Demonstration

•From Compton effect equation, 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 main reasoning that 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.

2.7 Compton Effect

Figure 2.24 Experimental confirmation of Compton scattering. The greater the scattering angle, the greater the wavelength change, in accord

with Compton effect equation.

2.7 Compton Effect

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 Solved Problems

1. (a) What are the energy and momentum of a photon of red light of

wavelength 650 nm? (b) What is the wavelength of a photon of

energy 2.40 eV?

2 Solved Problems

2. The maximum wavelength for photoelectric emission in tungsten is

230 nm. What wavelength of light must be used in order for

electrons with a maximum energy of 1.5 eV to be ejected?

2 Solved Problems

3. The work function for tungsten metal is 4.52 eV. (a) What is the

cutoff wavelength λ

for tungsten? (b) What is the maximum kinetic

energy of the electrons when radiation of wavelength 198 nm is

used? (c) What is the stopping potential in this case?

2 Solved Problems

4. A laser beam with an intensity of 120W/m

(roughly that of a small

helium-neon laser) is incident on a surface of sodium. It takes a

minimum energy of 2.3 eV to release an electron from sodium (the

work function φ of sodium). Assuming the electron to be confined

to an area of radius equal to that of a sodium atom (0.10 nm), how

long will it take for the surface to absorb enough energy to release

an electron?

2 Solved Problems

5. Electrons are accelerated in television tubes through potential

differences of about 10 kV. Find the highest frequency of the

electromagnetic waves emitted when these electrons strike the

screen of the tube. What kind of waves are these?

2 Solved Problems

6. A single crystal of table salt (NaCl) is irradiated with a beam of X

rays of wavelength 0.250 nm, and the first Bragg reflection is

observed at an angle of 26.3

. What is the atomic spacing of NaCl?

2 Solved Problems

7. X rays of wavelength 0.2400 nm are Compton-scattered, and the

scattered beam is observed at an angle of 60.0◦ relative to the

incident beam. Find: (a) the wavelength of the scattered X rays, (b)

the energy of the scattered X-ray photons, and (c) the kinetic energy

of the scattered electrons

2 Solved Problems

8. Gamma rays of energy 0.662 MeV are Compton scattered.

a. What is the energy of the scattered photon observed at a

scattering angle of 60

b. What is the kinetic energy of the scattered electrons?

Chapter 2 Particle Properties of Waves

Additional Materials

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.

2.8 Pair Production

•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.

(Pair annihilation)

2.8 Pair Production: 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.

2.8 Pair Production: Photon Absorption

•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).