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TIPLER P A and LLEWELLYN R A
Modern Physics
(Freeman 6e, 2012)
Chapter 03 - Compton Effect
.
3-4 X Rays and the Compton Effect
(a)
(b)
141
Kα
8
L series
3
6
I(λ) (relative)
I (λ) (relative)
7
K series
5
4
2
Kβ
3
V = 80 kV
2
1
1
0
0
λm
0.2
0.4
λm
0.6
0.8 1.0
λ, Å
1.2
1.4
V = 40 kV
1.6
V = 35 kV
0
0
0.2
0.4
λm
0.6 0.8
λ, Å
1.0
1.2
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.
becomes eV  hf  hc>lm or lm  hc>eV  1.2407  106 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 function 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 Effect
It 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 scattering 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 relation 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
TIPLER_03_119-152hr2.indd 141
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.
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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 - l 1 =
h
11 - cos u2
mc
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 wavelength of the electron. Its value is
Molybdenum
K α line
primary
(a)
Scattered by
graphite at
45°
(b)
Scattered
at 90°
(c)
135°
(d )
6°30�
7°
7°30�
Angle from calcite
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.
TIPLER_03_119-152hr2.indd 142
lc =
h
hc
1.24 * 103 eV # nm
=
=
= 0.00243 nm
2
mc
mc
5.11 * 105 eV
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 generally 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 scattered 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 of
Dl  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 electromagnetic 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 matter and a wave theory to describe the interference and diffraction of electromagnetic radiation. We will discuss this so-called wave-particle duality in more detail
in Chapter 5.
S1
R
Defining
slit
Calcite
S2
crystal
φ
Shutter
X-ray tube
(Mo target)
Bragg
spectrometer
Ionization
chamber
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.
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3-4 X Rays and the Compton Effect
143
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.]
More
More
erivation of Compton’s Equation, applying conservation of
D
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.
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 scattering 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:
h
11 - cos u2
mc
= 0.0024311 - cos 1202
l2 - l1 = l =
TIPLER_03_119-152hr3.indd 143
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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
= 0.015
l1
3. Combining these yields
0.0024311 - cos 1202
Dl
=
0.015
0.015
= 0.243 nm
l1 =
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 Comment
In 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  1019 C
3. Blackbody radiation
Stefan-Boltzmann law
Wein’s displacement law
Planck’s radiation law
Planck’s constant
4. Photoelectric effect
5. Compton effect
6. Photon-matter interaction
TIPLER_03_119-152hr2.indd 144
R  sT 4
3-4
lmT  2.898  103 m # K
-5
8phcl e hc>lkT - 1
h  6.626  1034 J # s
u(l) =
eV0  hf  
l2 - l1 =
h
11 - cos u2 mc
3-5
3-18
3-19
3-21
3-25
The (1) photoelectric effect, (2) the Compton effect, and (3) pair
production are the three ways of interaction.
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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, Neutrons, 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 Fundamental Physical Constants,” Reviews of Modern Physics 80,
633–730 (April 2008).
Richtmyer, F. K., E. H. Kennard, and J. N. Cooper, Introduction to Modern Physics, 6th ed., McGraw-Hill,
New York, 1969. This excellent text was originally
published in 1928, intended as a survey course for graduate 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 Electron, 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 particle itself by Lorentz.
3. Joseph J. Thomson (1856–1940). English physicist and
director, for more than 30 years, of the Cavendish Laboratory, 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 electrostatic 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 produced in outer space.
6. R. A. Millikan, Physical Review, 32, 349 (1911).
TIPLER_03_119-152hr2.indd 145
7. Mohr, P. J., B. N. Taylor, and D. B. Newell, “The Fundamental 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 predecessor 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), AustralianEnglish 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.
8/22/11 11:33
MORE CHAPTER 3, #1
Derivation of
Compton’s Equation
Let 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 =
hf1
E1
h
=
=
c
c
1
p2 =
hf2
E2
h
=
=
c
c
2
and
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
p 2e = p 21 + p 22 - 2p1 # p2
= p 21 + p 22 - 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 collision, the energy of the electron is 1E 20 + p 2e c 2 2 1>2.
m
E1 = hf1
p1 = h/λ 1
1
pe = –– E 2 – E02
c
φ
θ
E2 = hf2
p2 = h/λ 2
10
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.
More Chapter 3
Conservation of energy gives
p1 c + E 0 = p2 c + 1E 20 + p 2e c 2 2 1>2
Transposing the term p2c and squaring, we obtain
E 20 + c 2 1p1 - p2 2 2 + 2cE 0 1p1 - p2 2 = E 20 + p 2e c 2
or
p 2e = p 21 + p 22 - 2p1 p2 +
2E 0 1p1 - p2 2
c
3-27
Eliminating p 2e between Equations 3-26 and 3-27, we obtain
E 0 1p1 - p2 2
= p1 p2 11 - cos 2
c
Multiplying each term by hc>p1 p2 E 0 and using = h>p, we obtain Compton’s
equation:
2 - 1 =
hc
hc
11 - cos 2 =
11 - cos 2
E0
mc 2
or
2 - 1 =
h
11 - cos 2
mc
3-25
11