TIPLER P A and LLEWELLYN R A

Modern Physics

(Freeman 6e, 2012)

Chapter 03 - Compton Effect

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3-4 X Rays and the Compton Effect 141

becomes eV hf hc>lm or lm hc>eV 1.2407 106 V 1 m 1.24 103 V 1 nm. Thus, the Duane-Hunt rule is explained by Planck’s quantum hypothesis. (Notice that the value of lm can be used to determine h>e.)

The continuous spectrum was understood as the result of the acceleration (i.e., “braking”) of the bombarding electrons in the strong electric fields of the target atoms. Maxwell’s equation predicted the continuous radiation. The real problem for classical physics was the sharp lines. The wavelengths of the sharp lines were a func-tion of the target element, the set for each element being always the same. But the sharp lines never appeared if V was such that lm was larger than the particular line, as can be seen from Figure 3-15a, where the shortest-wavelength group disappears when V is reduced from 80 keV to 40 keV so that lm becomes larger. The origin of the sharp lines was a mystery that had to await the discovery of the nuclear atom. We will explain them in Chapter 4.

Compton EffectIt had been observed that scattered x rays were “softer” than those in the incident beam, that is, were absorbed more readily. Compton16 pointed out that if the scat-tering process were considered a “collision” between a photon of energy hf1 (and momentum hf1>c) and an electron, the recoiling electron would absorb part of the incident photon’s energy. The energy hf2 of the scattered photon would therefore be less than the incident one and thus of lower frequency f2 and momentum hf2>c. (The fact that electromagnetic radiation of energy E carried momentum E>c was known from classical theory and from experiments of Nichols and Hull in 1903. This rela-tion is also consistent with the relativistic expression E 2 p2c2 + (mc2)2 for a particle with zero rest energy.) Compton applied the laws of conservation of momentum and energy in their relativistic form (see Chapter 2) to the collision of a photon with an isolated electron to obtain the change in the wavelength l2 l1 of the photon as a

Well-known applications of x rays are medical and dental x rays (both diagnostic and treatment) and industrial x ray inspection of welds and castings. Perhaps not so well known is their use in determining the structure of crystals, identifying black holes in the cosmos, and “seeing” the folded shapes of proteins in biological materials.

Figure 3-15 (a) x-ray spectra from tungsten at two accelerating voltages and (b) from molybdenum at one. The names of the line series (K and L) are historical and explained in Chapter 4. The L-series lines for molybdenum (not shown) are at about 0.5 nm (5 Å). The cutoff wavelength lm is independent of the target element and is related to the voltage on the x-ray tube V by lm hc>eV. The wavelengths of the lines are characteristic of the element.

(a)I(

λ) (

rela

tive)

6

7

8

5

4

3

2

1

00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

λ, Åλmλm λm

K series

L series

V = 80 kV

V = 40 kV

I(λ)

(re

lativ

e)

3

2

1

00 0.2 0.4 0.6 0.8 1.0 1.2

λ, Å

V = 35 kV

Kβ

Kα(b)

TIPLER_03_119-152hr2.indd 141 8/22/11 11:33 AM

142 Chapter 3 Quantization of Charge, Light, and Energy

function of the scattering angle u. The result, called Compton’s equation and derived in a More section on the home page, is

l2 - l1 =h

mc11 - cos u2 3-25

The change in wavelength is thus predicted to be independent of the original wavelength. The quantity h>mc has the dimensions of length and is called the Compton wave-length of the electron. Its value is

lc =h

mc=

hc

mc2 =1.24 * 103 eV # nm

5.11 * 105 eV= 0.00243 nm

Because l2 l1 is small, it is difficult to observe unless l1 is very small so that the fractional change (l2 l1)>l1 is appreciable. For this reason Compton effect is gen-erally only observed for x rays and gamma radiation.

Compton verified his result experimentally using the characteristic x-ray line of wavelength 0.0711 nm from molybdenum for the incident monochromatic photons and scattering these photons from electrons in graphite. The wavelength of the scat-tered photons was measured using a Bragg crystal spectrometer. His experimental arrangement is shown in Figure 3-16; Figure 3-17 shows his results. The first peak at each scattering angle corresponds to scattering with no shift in the wavelength due to scattering by the inner electrons of carbon. Since these are tightly bound to the atom, it is the entire atom that recoils rather than the individual electrons. The expected shift in this case is given by Equation 3-25, with m being the mass of the atom, which is about 104 times that of the electron; thus, this shift is negligible. The variation ofDl l2 l1 with u was found to that predicted by Equation 3-25.

We have seen in this and the preceding two sections that the interaction of elec-tromagnetic radiation with matter is a discrete interaction that occurs at the atomic level. It is perhaps curious that after so many years of debate about the nature of light, we now find that we must have both a particle (i.e., quantum) theory to describe in detail the energy exchange between electromagnetic radiation and mat-ter and a wave theory to describe the interference and diffraction of electromag-netic radiation. We will discuss this so-called wave-particle duality in more detail in Chapter 5.

φ

X-ray tube(Mo target)

R

S1

DefiningslitS2

Calcitecrystal

Braggspectrometer

Ionizationchamber

Shutter

Figure 3-16 Schematic sketch of Compton’s apparatus. x rays from the tube strike the carbon block R and are scattered into a Bragg-type crystal spectrometer. In this diagram, the scattering angle is 30°. The beam was defined by slits S1 and S2. Although the entire spectrum is being scattered by R, the spectrometer scanned the region around the Ka line of molybdenum.

Figure 3-17 Intensity versus wavelength for Compton scattering at several angles. The left peak in each case results from photons of the original wavelength that are scattered by tightly bound electrons, which have an effective mass equal to that of the atom. The separation in wavelength of the peaks is given by Equation 3-25. The horizontal scale used by Compton “angle from calcite” refers to the calcite analyzing crystal in Figure 3-16.

(a)

(c)

Scatteredat 90°

(b)

(d )

135°

MolybdenumKα lineprimary

Scattered bygraphite at45°

6°30� 7°30�7°Angle from calcite

TIPLER_03_119-152hr2.indd 142 8/22/11 11:33 AM

3-4 X Rays and the Compton Effect 143

More Derivation of Compton’s Equation, applying conservation of energy and momentum to the relativistic collision of a photon and an electron, is included on the home page: www.whfreeman.com /tiplermodernphysics6e. See also Equations 3-26 and 3-27 and Figure 3-18 here.

More

Questions

6. Why is it extremely difficult to observe the Compton effect using visible light?

7. Why is the Compton effect unimportant in the transmission of television and radio waves? How many Compton scatterings would a typical FM signal have before its wavelengths were shifted by 0.01 percent?

EXAMPLE 3-9 Compton Effect In a particular Compton scattering experiment it is found that the incident wavelength l1 is shifted by 1.5 percent when the scatter-ing angle u 120°. (a) What is the value of l1? (b) What will be the wavelength l2 of the shifted photon when the scattering angle is 75°?

SOLUTION 1. For question (a), the value of l1 is found from Equation 3-25:

l2 - l1 = l =h

mc11 - cos u2

= 0.0024311 - cos 1202

Arthur Compton. After discovering the Compton effect, he became a world traveler seeking an explanation for cosmic rays. He ultimately showed that their intensity varied with latitude, indicating an interaction with Earth’s magnetic field, and thus proved that they were charged particles. [Courtesy of American Institute of Physics, Niels Bohr Library.]

TIPLER_03_119-152hr3.indd 143 8/31/11 10:13 AM

144 Chapter 3 Quantization of Charge, Light, and Energy

2. That the scattered wavelength l2 is shifted by 1.5 percent from l1 means that

Dl

l1= 0.015

3. Combining these yields

l1 =Dl

0.015=

0.0024311 - cos 12020.015

= 0.243 nm

4. Question (b) is also solved with the aid of Equation 3-25, rearranged as

l2 = l1 + 0.0024311 - cos u2 5. Substituting u 75 and l1 from above yields

l2 = 0.243 + 0.0024311 - cos 752 = 0.243 + 0.002 = 0.245 nm

A Final CommentIn this chapter together with Section 2-4 of the previous chapter we have introduced and discussed at some length the three primary ways by which photons interact with matter: (1) the photoelectric effect, (2) the Compton effect, and (3) pair production. As we proceed with our explorations of modern physics throughout the remainder of the book, we will have many occasions to apply what we have learned here to aid in our understanding of a myriad of phenomena, ranging from atomic structure to the fusion “furnaces” of the stars.

Summary TOPIC RELEVANT EQUATIONS AND REMARKS

1. J. J. Thomson’s experiment Thomson’s measurements with cathode rays showed that the same particle (the electron), with e>m about 2000 times that of ionized hydrogen, exists in all elements.

2. Quantization of electric charge e 1.60217653 1019 C

3. Blackbody radiation

Stefan-Boltzmann law

Wein’s displacement law

Planck’s radiation law

Planck’s constant

R sT 4 3-4

lmT 2.898 103 m # K 3-5

u(l) =8phcl-5

ehc>lkT - 1 3-18

h 6.626 1034 J # s 3-19

4. Photoelectric effect eV0 hf 3-21

5. Compton effectl2 - l1 =

h

mc11 - cos u2 3-25

6. Photon-matter interaction The (1) photoelectric effect, (2) the Compton effect, and (3) pair production are the three ways of interaction.

TIPLER_03_119-152hr2.indd 144 8/22/11 11:33 AM

Notes 145

General References The following references are written at a level appropriate for the readers of this book.

Millikan, R. A., Electrons (1 and ) Protons, Photons, Neu-trons, Mesotrons, and Cosmic Rays, 2d ed., University of Chicago Press, Chicago, 1947. This book on modern physics by one of the great experimentalists of his time contains fascinating, detailed descriptions of Millikan’s oil-drop experiment and his verification of the Einstein photoelectric-effect equation.

Mohr, P. J., B. N. Taylor, and D. B. Newell, “The Fundamen-tal Physical Constants,” Reviews of Modern Physics 80, 633–730 (April 2008).

Richtmyer, F. K., E. H. Kennard, and J. N. Cooper, Intro-duction to Modern Physics, 6th ed., McGraw-Hill,New York, 1969. This excellent text was originally

published in 1928, intended as a survey course for grad-uate students.

Shamos, M. H. (ed.), Great Experiments in Physics, Holt, Rinehart, & Winston, New York, 1962. This book contains 25 original papers and extensive editorial comment. Of particular interest for this chapter are papers by Faraday, Hertz, Roentgen, J. J. Thomson, Einstein (photoelectric effect), Millikan, Planck, and Compton.

Thomson, G. P., J. J. Thomson, Discoverer of the Elec-tron, Doubleday/Anchor, Garden City, NY, 1964. An interesting study of J. J. Thomson by his son, also a physicist.

Weart, S. R., Selected Papers of Great American Physicists, American Institute of Physics, New York, 1976. The bicentennial commemorative volume of the American Physical Society.

Notes 1. Democritus (about 470 b.c. to about 380 b.c.). Among

his other modern-sounding ideas were the suggestion that the Milky Way is a vast conglomeration of stars and that the Moon, like Earth, has mountains and valleys.

2. G. J. Stoney (1826–1911). An Irish physicist who first called the fundamental unit of charge the electron. After Thomson discovered the particle that carried the charge, the name was transferred from the quantity of charge to the par-ticle itself by Lorentz.

3. Joseph J. Thomson (1856–1940). English physicist and director, for more than 30 years, of the Cavendish Labora-tory, the first laboratory in the world established expressly for research in physics. He was awarded the Nobel Prize in Physics in 1906 for his work on the electron. Seven of his research assistants also won Nobel Prizes.

4. There had been much early confusion about the nature of cathode rays due to the failure of Heinrich Hertz in 1883 to observe any deflection of the rays in an electric field. The failure was later found to be the result of ionization of the gas in the tube; the ions quickly neutralized the charges on the deflecting plates so that there was actually no electric field between the plates. With better vacuum technology in 1897, Thomson was able to work at lower pressure and observe elec-trostatic deflection.

5. R. A. Millikan, Philosophical Magazine (6), 19,209 (1910). Millikan, who held the first physics Ph.D. awarded by Columbia University, was one of the most accomplished experimentalists of his time. He received the Nobel Prize in Physics in 1923 for the measurement of the electron’s charge. Also among his many contributions, he coined the phrase cosmic rays to describe radiation pro-duced in outer space.

6. R. A. Millikan, Physical Review, 32, 349 (1911).

7. Mohr, P. J., B. N. Taylor, and D. B. Newell, “The Funda-mental Physical Constants,” Reviews of Modern Physics 80, 633–730 (April 2008).

8. See pp. 135–137 of F. K. Richtmyer, E. H. Kennard, and J. N. Cooper (1969).

9. John W. S. Rayleigh 1842–1919. English physicist, almost invariably referred to by the title he inherited from his father. He was Maxwell’s successor and Thomson’s prede-cessor as director of the Cavendish Laboratory.10. Max K. E. L. Planck (1858–1947). Most of his career was spent at the University of Berlin. In his later years his renown in the world of science was probably second only to that of Einstein.11. Heinrich R. Hertz (1857–1894). German physicist, student of Helmholtz. He was the discoverer of electromagnetic “radio” waves, later developed for practical communication by Marconi.12. H. Hertz, Annalen der Physik, 31, 983 (1887).13. A. Einstein, Annalen der Physik, 17, 144 (1905).14. A translation of this paper can be found in E. C. Watson, American Journal of Physics, 13, 284 (1945), and in Shamos (1962). Roentgen (1845–1923) was honored in 1901 with the first Nobel Prize in Physics for his discovery of x rays.15. William Lawrence Bragg (1890–1971), Australian-English physicist. The work that Bragg, an infant prodigy, performed on x-ray diffraction with his father, William Henry Bragg (1862–1942), earned for them both the Nobel Prize in Physics in 1915, the only father-son team to be so honored thus far. In 1938 W. L. Bragg became director of the Cavendish Laboratory, succeeding Rutherford.16. Arthur H. Compton (1892–1962), American physicist. It was Compton who suggested the name photon for the light quantum. His discovery and explanation of the Compton effect earned him a share of the Nobel Prize in Physics in 1927.

TIPLER_03_119-152hr2.indd 145 8/22/11 11:33 AM

10

MORE CHAPTER 3, #1

Derivation of Compton’s EquationLet �1 and �2 be the wavelengths of the incident and scattered x rays, respectively, as shown in Figure 3-18. The corresponding momenta are

p1 =E1

c=

hf1

c=

h

�1

and

p2 =E2

c=

hf2

c=

h

�2

using f� � c. Since Compton used the K� line of molybdenum (� � 0.0711 nm; see Figure 3-15b), the energy of the incident x ray (17.4 keV) is much greater than the binding energy of the valence electrons in the carbon-scattering block (about 11 eV); therefore, the carbon electrons can be considered to be free.

Conservation of momentum gives

p1 = p2 + pe

or

p2e = p2

1 + p22 - 2p1

# p2

= p21 + p2

2 - 2p1p2 cos � 3-26

where pe is the momentum of the electron after the collision and � is the scattering angle of the photon, measured as shown in Figure 3-18. The energy of the electron before the collision is simply its rest energy E0 � mc2 (see Chapter 2). After the colli-sion, the energy of the electron is 1E2

0 + p2ec

221>2.

FIGURE 3-18 The scattering of x rays can be treated as a collision of a photon of initial momentum h/�1 and a free electron. Using conservation of momentum and energy, the momentum of the scattered photon h/�2 can be related to the initial momentum, the electron mass, and the scattering angle. The resulting Compton equation for the change in the wavelength of the x ray is Equation 3-25.

E1 = hf1

E2 = hf2

p1 = h/λ1

p2 = h /λ2

φ

θ

mpe = E2 – E0

21––c

More Chapter 3 11

Conservation of energy gives

p1 c + E0 = p2 c + 1E20 + p2

e c221>2 Transposing the term p2c and squaring, we obtain

E20 + c21p1 - p222 + 2cE01p1 - p22 = E2

0 + p2e c2

or

p2e = p2

1 + p22 - 2p1 p2 +

2E01p1 - p22c

3-27

Eliminating p2e between Equations 3-26 and 3-27, we obtain

E01p1 - p22

c= p1 p211 - cos �2

Multiplying each term by hc>p1 p2 E0 and using � = h>p, we obtain Compton’s equation:

�2 - �1 =hc

E011 - cos �2 =

hc

mc2 11 - cos �2 or

�2 - �1 =h

mc11 - cos �2 3-25