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The Evidence for the Photon
The Atomic Nature of Matter and the Elementary Structure of the Atom
The first evidence that nature is quantized arose from the realization that there
were only a small number of fundamental elements that combine in fixed ratios to
produce all the know substances of nature. In the latter 19th century physicist applied
this concept of the atomic nature of matter to a gas confined in a volume and were able to
derive the ideal gas law using only a few very simple assumptions. These and other
observations were strong evidence that the world in which we live is quantized. As we
shall see in what follows, the quantum nature of things is not restricted just to physical
material, but manifests itself in many different areas - quantized energy levels, quantized
spatial orientations, etc. Many of the experiments performed in the latter 19th and early
20 century demonstrated this quantum nature of our world. It is our task to examine some
of the major experiments from that period of time.
J. J. Thompson' Experiments With Electrons
Evidence for Photons
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Blackbody Radiation
Max Planck's explanation of blackbody radiation presented at the German Physical
Society on December 14, 1900 may be considered the beginning of quantum theory. In
this section we will briefly outline the theory of blackbody radiation and the events which
led up to Planck's historic paper.
All objects radiate and absorb energy in the form of electromagnetic radiation. An
object in equilibrium with its surroundings must necessarily emit and absorb equal
amounts of energy. If this were not so, the object would either heat up as other objects
cooled off, or vice versa! This is basically a statement of Kirchhoff's law of absorption
and emission.
We define V (X ), the radiated power (or radiant flux density) from a heated source, as
the total amount of energy radiated in all wavelengths from the source per unit time per
unit surface area of the source. In 1879 Joseph Stephan determined experimentally that
V (X ) = -98=>+8> ‚ X %
where X is the absolute temperature. Today we express this equation in the form
V X œ %5X %
where % is the emissivity and depends upon the emitting surface, and where 5 is a
universal constant (now known as the Stephan-Boltzmann constant) given by:
5 = 5.67 x 10-8 W/m# -K%
Objects with an emissivity equal to unity are known as blackbodies. This is because an
object in thermal equilibrium must emit and absorb equal amounts of radiation. Thus, %
is also the absorption coefficient, and if the absorption coefficient is equal to unity, then
the object must absorb all the radiation incident upon it, so it is a "black" body!
EXERCISE: Assume that the sun's surface temperature is 6000 K and determine the
equilibrium temperature of the earth, assuming that both the sun and the earth are ideal
emitters and absorbers (i.e., % = 1).
The wavelength or frequency of the energy emitted depends upon the temperature of
the object. Thus
V (X ) = ' VÐ/ ,X ) d/ ,
∞
o
where V (/ ,X ), the spectral radiant flux density, is the amount of energy radiated per unit
time per unit area in the frequency interval between / and / + d/ . The distribution
function for radiated energy as a function of frequency was first determined
experimentally by Lummer and Pringsheim in 1899, and is shown in the following figure.
Evidence for Photons
3
Blackbody Radiation Curves for Different Temperatures.
40
Spectral Intensity (arbitrary units)
35
30
25
20
3000K
15
2000K
10
5
1000K
0
0
5
10
15
20
25
30
35
frequency (arbitrary units)
40
45
50
As this figure shows, when an object is heated up, the total amount of energy radiated
from the object increases (the area under the curve increases), and the peak in the
radiation curve shifts toward higher frequencies (or shorter wavelengths, since -/ œ c
Ê - º /" ). Thus, an object at room temperature radiates in the far infra-red region of
the spectrum (invisible to the naked eye), while a much hotter object might glow red-hot,
and a still hotter object might glow white-hot (i.e, the radiated light would contain much
more of the visible spectrum). [This simple observation explains why “red” stars are
cooler then “blue” stars.] It was demonstrated by Wein that the frequency at which
V (/ ,X ) is a maximum is proportional to the absolute temperature, or
/7+B º X
This is called Wein's displacement law and can also be written in the form
-7+B º
"
X
Based upon classical thermodynamics, Wein argued that V (/ ,T) should have the form
V (/ ,X ) = / $ J (/ /X )
where J (/ /X ) was some function dependent only on the ratio / ÎX . Wein showed that
the empirical formula was of the form
V (/ ,X ) = +/ $ exp[-, / /5X ],
Evidence for Photons
4
where + and , are arbitrary constants, and where 5 is the Boltzmann constant. This
expression for the equation indeed gives Wein's displacement law and seemed to fit the
existing experimental data (for low frequencies) reasonably well.
EXERCISE: Beginning with Wein's expression for the spectral radiant flux density
derive the Wein displacement law and the Stephan-Boltzmann equation.
An ideal blackbody can be constructed using an enclosed cavity which has only a
small hole open to the outside. Any light which falls on this hole will enter the cavity
and be continually reflected inside the cavity until it is eventually absorbed, because
there is little likelihood that the light will be reflected back out through the small
opening. This is truly a blackbody (a black hole). But if the object containing the cavity
is heated, the opening will also act as an emitter of radiation, and will emit a blackbody
spectrum. Now the amount of energy which exits the hole per unit area of hole per unit
time in a particular frequency range is given by V (/ ,X ) and must be related to the total
energy which is contained in the cavity. In fact, the energy density ?(/ ,X ) times the
speed of light (the rate at which energy will leave the opening) has units of energy per
unit area per unit time so that we expect
V (/ ,X ) º - ?(/ ,X )
ÒIt is in fact given by V / ß X œ %- ? / ß X ÓÞ This means that the energy density within a
cavity radiator should follow Wein's law and be given by
? / ß X º +/ $ exp[, / /5X ]
[Note: The arbitrary constant + in this equation is not identical with the one in the
previous equation. The constant - and others have been absorbed into this +.]
II. THEORETICAL ATTEMPTS TO EXPLAIN BLACKBODY RADIATION
In an attempt to derive theoretically the energy density distribution function ?(/ ,X ),
Rayleigh and Jeans assumed that the interior of a blackbody cavity was made up of
perfectly reflecting material (this poses no theoretical problem, since all blackbodies can
be shown to be equivalent). Thus, the electromagnetic energy reflected back and forth
inside the cavity must be treated something like standing waves (with nodes at both
ends). The trick is to determine the number of different ways standing waves can be set
up inside the cavity for a given frequency interval (between / and /  d/ ), and to
determine the average energy density that such a wave can have.
Classical arguments indicate tha the average energy stored in a cavity is equal to
5X times the number of possible modes of oscillation of the waves trapped in the cavity
ØIÙ œ ØIÙÎ79./ ‚ 8?7,/< 90 79./=
Now the number of modes of oscillation depends upon the frequency of the wave trapped
in the cavity. Classical arguments indicated tha the number of modes of a transverse
Evidence for Photons
5
wave with a frequency between / and /  . / is given by
)1/ #
./
-$
. Átransverse œ Z
The average energy in a given frequency range, then, is given by
ØE(/ )Ù d/ œ Z
)1/ #
5X . /
-$
and the average energy density (energy per unit volume) in that frequency range is given
by
? / . / œ 5X
)1/ #
./
-$
A Comparison Of Experiment And Theory
Wein's emperical law for the energy density of a blackbody radiator
? / ß X . / º +/ $ exp[, / /5X ] . /
is not the same as the energy density derived by Rayleigh and Jeans, based upon classical
argurments. In fact, their equation
? / ,X . / œ 5X
)1 / #
./
-$
appears to be quite different. Plotting these two functions with respect to the frequency
(see below) we see that the two functions agree for low frequencies, but at high
frequencies, the Rayleigh-Jeans law fails badly.
Energy Density as a Function of Frequency
5
Energy Density (arbitrary units)
4
Rayleigh-Jean's Law
3
2
Wein's Law
1
0
0
2
4
6
Frequency (arbitrary units)
8
10
Evidence for Photons
6
Planck recognized a connection between the Rayleigh-Jeans law and Wein's law.
If we write Wein's law in the form
? /ß X œ
+/ $
expc, / Î5X d
and expand the exponential using the Taylor series, we obtain
? /ß X œ
’" 
+/ $
,/
5X
,/ ‰
 ˆ 5X
 â“
#
In the limit as , / Î5X becomes small enough to ignore terms higher than first order, we
can see that this equation is almost exactly the Rayleigh-Jeans law - except for the factor
of one " in the denominator. As a result, Planck proposed that the correct form of the
blackbody radiation equation should be
+/ $
? /ß X œ
expc, / Î5X d  "
which has the form of the Rayleigh-Jeans law for low frequencies (long wavelengths) and
the form of Wein's law for high frequencies (short wavelengths). To make this equation
have the exact form of the Rayleigh-Jeans law at low frequencies, we must choose the
emperical constants + and , in Wein's equation appropriatelyÞ In the limit of small
, / Î5X , we have
? /ß X œ
+/ $
+/ $
+
p ,/ œ / # 5X
expc,/ Î5X d  "
,
5X
which means that
+
)1
œ $
,
This gives as the proper form of the Blackbody radiation equation
? / ß X ./ œ
)1/ #
2/
./
$
- expc2 / Î5X d  "
where we have replaced , with 2 so that it is in the traditional form of Planck's equation.
We can determine the value of the constant , by integrating the energy density over all
frequecies.
You should notice that 2 / must have units of energy (the same units as 5X ) and
that the term
2/
expc2 / Î5X d  "
has taken the place of 5X in the Rayleigh-Jean's law. You will recall that 5X was the
classically predicted average energy of an oscillator. Thus, Planck's equation implies that
Evidence for Photons
7
the correct form for the average energy of an oscillator is given by
ØIÙ œ
2/
expc2 / Î5X d  "
Notice that this expression for the energy is identical to the classical expression in the
limit as 2/ goes to zero! In fact, Planck found that the classical calculation of the
average energy could be formulated to give the result above, with 2/ being the smallest
increment of energy. Classically, this smallest increment is zero (i.e., we let 2 p ! in the
expression for the energy, 2/ ), and we obtain the classical value for the average energy,
5X . The experimentally measured blackbody spectrum, however, demands that 2 cannot
go to zero! Thus, we find that nature has a lower limit on energy increments: energy
cannot be subdivided into units smaller than 2/ !
Planck's constant is obviously related to the Stephan-Boltzmann constant. If we
write
V /ß X œ
#1/ #
2/
? /ß X œ #
2
/
Î5X
%
- /
"
we can find the relationship between the Stephan-Boltzman constant 5 and Planck's
constant 2 , by integrating over all frequencies, giving
∞
V (X ) = (
o
∞
VÐ/ ,X ) d/ œ (
o
#1/ #
2/
. / œ 5X %
#
2
Î5X
/
- /
"
The value of Planck's constant turns out to be
2 œ 'Þ'#'!(' ‚ "!$% J † s
œ %Þ"$&''* ‚ "!"& eV † s
Evidence for Photons
8
The Photoelectric Effect
The photoelectric effect was first observed in 1887 by Hertz as he was
experimenting with radio waves. Hertz discovered that a spark could be more easily
initiated at a spark gap in the presence of other sparks. As he studied this phenomena
further, he found that the enhancement of the sparks was due to ultraviolet light, that the
effect was more pronounced if the terminals were clean and smooth, and that the negative
terminals were the more sensitive to the incident light.
G
Soon after Hertz' initial observations, others began to study this phenomenon. A
schematic of a typical vacuum tube used to study the photoelectric effect is shown in the
diagram above, and the results of some of these experimental studies are listed below.
[Notice the continuity of the names of persons who were involved in the research and the
dates when the research was reported.]
1.
2.
3.
4.
5.
6.
7.
8.
The photoelectric effect involves negatively charged particles. (Hallwachs, 1889)
These charged particles are emitted forcibly by the light. (Hallwachs, Elster, and
Geitel, 1889)
A close relationship exists between the contact potential (or electronegativity) of the
metal and its sensitivity to light (photosensitivity). The more electropositive a metal
the more sensitive it is to longer wavelengths. (Elster and Geitel, 1889)
The photocurrent is proportional to the intensity of the light. (Elster and Geitel,
1891)
The emitted particle was determined to be the electron. (Thompson and Lenard,
1899)
The kinetic energy of an emitted electron is independent of the intensity of the
incident light, but the number of emitted electrons is proportional to the light
intensity. (Lenard, 1902)
The maximum kinetic energy of the emitted electrons is greater when the light
incident upon the metal has a shorter wavelength. No electrons whatsoever are
emitted if the wavelength is longer than some maximum threshold value. (Lenard,
1902)
The photoelectrons were observed to be emitted from the metal surfaces with no
appreciable time delay after the source was turned on.
Evidence for Photons
9
Although Newton had first postulated that light was corpuscular in nature (i.e., that
light consisted of small 'particles'), Young's double slit experiment had firmly
established that light was a wave: it clearly exhibited interference phenomena. By the the
early 1900's, the mathematical models used to describe waves and wave interference had
been successful in explaining many problems involving light. However, many of the
experimental observations concerning the photoelectric effect could in no way be
explained by the classical wave model of light.
One particular aspect of classical wave theory is worth noting here. The intensity of
light (a measure of the amount of energy which is incident upon a surface per unit time)
is calculated by taking the square of the absolute magnitude of the amplitude of the
electromagnetic wave at a point in space. This wave amplitude might consist of the
summation of several different waves coming from different sources (which is how one
explains the phenomena of interference), but the intensity of light at the point of interest
can only be determined by taking the square of the absolute magnitude of the resultant
electromagnetic wave. As a simple example, consider a sine wave as a model for light.
We know that the time average of the sine wave is zero if the average is taken over a
complete period. However, the time average of the square of the sine wave will not be
zero, and the classical model of light would imply that if light is incident upon a surface
for a long time, then the energy from the light wave which is imparted to the surface will
increase with time.
However, the observations of the photoelectric effect (in particular 6 and 8) would
seem to disagree with this picture. One would think that if the light source were turned
on, the energy from the source would continually be added to the metal surface until
enough energy was added to eject an electron. This would imply some time delay
between turning on the source and the ejection of an electron. But there was no
measurable time delay if an electron were ejected at all. And the energy of the ejected
electron was dependent, not on how long the light source had been on, but rather on the
frequency of the incident light, a condition which was in no way indicated by the
classical theory. Rather classical theory would predict that the photoelectric effect
should occur for any frequency as long as the source were on long enough.
The pertinent observations concerning the photoelectric effect are summarized by
two diagrams below. The first diagram plots the photocurrent (the current of ejected
electrons) as a function of "retarding potential" for two different intensities of the source
(the sources both being monochromatic sources of the same wavelength). Each electron
which reaches the collector plate has an energy of "# 7/ @# , the most energetic being the
ones which reach the plate at Z9 . Thus, we have
"
#
# 7/ @7+B
œ /Z9
which is independent of the intensity of the source, but which does depend upon the
incident radiation frequency.
Evidence for Photons
10
Ia
Ib
Vo
0
Stopping Potential (V)
If we plot the cut-off potential for a given metal as a function of the frequency of the
incident light, we obtain the following diagram, where the results for two different metals
are plotted together. Here we see that both curves exhibit the same slope, but different
intercepts, depending upon the photosensitive metal.
Metal 1
Metal 2
0
Light Frequency
When Planck's paper on the quantum explanation of blackbody radiation came out
in 1903, Einstein immediately saw that this idea of quantized energies could be used to
explain the phenomena of the photoelectric effect. He postulated that light was made up
of "photons" each of which has a "quantized" energy, % œ 2 / , and that the intensity of
light was given by 82 / , where 8 is the number of photons present. This hypothesis
would immediately indicate that the energy of a photon is directly related to the
frequency of the incident light, and that the photocurrent which is observed should be a
function of the number of photons (the intensity of the source) which arrive at the
photosensitive metal. In this hypothesis, the energy of a single photon, 2/ would be
given to a single electron, so that the electron would acquire a kinetic energy which could
be determined by the equation
2/ œ
"
#
7/ @7+B
[
#
where [ is the work function for a particular photosensitive substance. Now, since the
maximum kinetic energy is equal to the electrostatic potential energy /Z9 , we can write
2 / œ /Z9  [
Evidence for Photons
11
or, solving for Z9 , we have
Z9 œ Œ
2
/
/
[
/
and we see that this model for the energy in an electromagnetic wave fits the
photoelectric effect data very well. The fact that the slope of the data fits the numerical
value of 2Î/ was first demonstrated by Millikan when he made an attempt to disprove
Einstein's hypothesis. Einstein's theoretical explanation of the photoelectric effect
probably did more to establish Planck's quantum theory than any other phenomena.
An interesting problem has now been introduced, however. A proper description of
light evidently requires us to somehow blend the idea that light can be described as a
wave but also as a discrete "particle". The resolution of this "problem" will take up a
considerable part of our study.
Evidence for Photons
12
X-rays
X-rays were first discovered in 1895 by Röntgen. In his first attempts to identify
these mysterious “rays” Röntgen discovered: 1) they could not be deflected by a magnet,
and so it seemed they were not charged particles; and 2) they exhibited no apparent
diffraction or refraction effects, and so it seemed they were not electromagnetic waves.
Because he had no idea what these rays were, Röntgen coined the name “x-ray”.
Although he did not initially detect any deflection of these x-rays by diffraction or
refraction effects, Röntgen reasoned that these rays were electromagnetic in origin, since
they were produced when electrons were accelerated toward a metal electrode. He
reasoned that the charged electrons were decelerated within the target metal, and the
decelerating charge would radiate energy. Indeed, further study of these x-rays indicated
that an x-ray beam was indeed slightly broadened when passed through an extremely
narrow slit. This seemed to indicate that the wavelength of these x-rays must be
extremely short.
Electromagnetic waves are diffracted more when they pass through openings
which are comparable in size to their wavelength. Since the wavelength of x-rays
appeared to be very short, one needed a diffraction grating with very small spacing. In
1912 Laue proposed using a crystal as a three-dimensional grating since the grating
spacing would be on the order of the atomic dimension. Soon after this, Friedrich and
Knipping were successful in obtaining a x-ray diffraction pattern. Their experiment
confirmed Röntgen's belief that the x-rays were indeed electromagnetic waves and also
confirmed the idea that the atoms in a crystal were arranged in a regular array.
Using crystal gratings, the spectral intensity of x-rays could be studied. These
studies indicated that there were two distinct patterns to x-ray emissions as seen in the
diagram below. The first type of x-ray emission are the relatively sharp x-ray lines which
depend upon the type of target used in the x-ray tube. These lines typically occur in pairs
as indicated in the figure. Here a Oα and O" pair are indicated. Second, a continuous
spectrum is observed which appears to have a short wavelength limit, -o . This part of the
x-ray spectrum is known as Bremsstrahlung radiation.
Evidence for Photons
13
I
Kα
Kβ
Bremsstrahlung
λο
λ
It was determined experimentally that the low wavelength limit to
Bremsstrahlung radiation varied inversely with the accelerating potential applied to the xray tube. Assuming, as did Röntgen, that x-radiation is produced when the bombarding
electrons decelerate near charged particles within the target, and that the radiation is in
the form of photons, we can argue that the energy of the photon is the energy given up by
the electron as it decelerates. The maximum energy which the electron has is the energy
given to it by the accelerating potential applied to the x-ray tube. That energy is given by
the electron charge multiplied by the accelerating potential. Thus the maximum energy
photon is given by
2 /7+B œ
2œ /Zo
-738
giving
-738 œ
2/Zo
in agreement with the observations. Thus, the observed spectrum of Bremsstrahlung
radiation also indicates that electromagnetic waves can best be described in terms of
photons of energy 2/ .
From his study of the x-ray emissions of different atomic species, Moseley found
a relationship between the wavelength or frequency of the Oα line and the atomic
number of the form
^ œE
"
 W œ F È /α  W
È -α
where ^ is the atomic number, -α and /α are the wavelength and frequency,
respectively, of the emitted Oα line, and where E, F , and W are constants. This
relationship is immediately derivable from Bohr's model for a hydrogenic atom.
Evidence for Photons
14
According to Bohr's model, the frequency of an emitted photon is given by
2/ œ 
^ # Io
^ # Io
"
"
#

”
• œ ^ Io ” #  w # •
#
w
8
8
8
8
For fixed values of 8 and 8w , we have
^ œ F È /α
which is pretty close to Moseley's equation.
If we realize that the transitions which we observe are arrising from electrons is
outer levels falling into the innermost level of the atom, we see that the nucleus of the
atom is somewhat “shielded” by electrons which may surround the nucleus. This
“shielding” of the nucleus by the innermost electrons effectively reduces the positive
electric charge of the nucleus by some amount (proportional to the number of electrons
shielding the nucleus?) which we write at ^  W , where W is the effective shielding
number.
It is found that for transition to the lowest energy level, W ¸ ". For transitions to
the next highest level, W ¸ 7.4.
Evidence for Photons
15
Compton Scattering of X-rays and Gamma Rays
As the study of x-rays continued, it was discovered that when x-rays scatter from
solids, the wavelength of the scattered x-ray is different from the original wavelength.
Experimentally, one would use an x-ray source with a strong Oα line of known
wavelenth. The x-rays from this source would be directed at a target and the wavelength
of the scattered x-rays would be analyzed using a crystal as a grating. It was found that
the shift in the wavelength was proportional to the scattering angle, according to the
equation
-w  -o º "  cos)
In 1925 Compton explained this wavelength shift and derived the Compton
equation
- w  -o œ
2"  cos)
7/ -
using the assumption that the energy of the electronmagentic wave was carried in a
quantized packet which we call the photon. To derive this equation Compton made use of
the fact that a photon must have momentum as described by Einstein's theory of special
relativity
I # œ :# - #  7o# - % œ :# - #
even though the rest mass of a photon is zero. Using the Einstein equation for the energy
of a photon, we obtain
I œ 2/ œ
2œ :-
from which we obtain the relation
:œ
2
-
Using this relationship and the assumption that photons act something like billiard
balls when they strike other objects, Compton applied the principles of conservation of
momentum and energy in a simple collision problem to derive his now famous equation.
To begin the analysis of Compton scattering, consider the diagram below. An
incident photon (wavelength -o Ñ collides with an electron which is initially at rest within
the target material. The scattered photon (wavelength -') moves off at an angle ) relative
to the direction of the incident photon, while the electron is given a kinetic energy X and
moves off at an angle 9 with respect to the direction of the incident photon.
Evidence for Photons
16
y
λ'
λο
θ
x
φ
T
We now apply the conservation of relativistic energy and momentum to this problem in
both the B and the C direction.
Before the collision, the energy and momentum of the incident photon is given by
2
-o
2I o œ 2 /o œ
œ :o -o
:o œ
also, the target electron has rest mass 7o - # , so the total energy before the collision is
given by
I3 œ
2 7o - # œ : o -  7 o - #
-o
After the collision, the energy of the system can be expressed by the equation
I0 œ
2 # ? 7o - # œ : w -  I /
-w
The conservation of energy requires that I3 œ I0 or
:o -  7o - # œ :w -  I/
from which we can derive the energy of the electron
I/ œ : o  : w -  7 o - #
The conservation of momentum requires that
:o œ :/ cos 9  :w cos )
Evidence for Photons
17
and
:/ sin 9 œ :w sin )
This leaves us with three equations and four unknows (),9 , I/ , I w ß where :/ and :w are
not independent unknowns). We cannot solve this uniquely, but we can eliminate the
dependence on the electron energy and the angle 9. To eliminate the angle 9, we write
the two momentum conservation equations
:/ cos 9 œ :o  :w cos )
:/ sin 9 œ :w sin )
Now squaring both equations and adding gives
#
:/# œ :o#  :w  #:o :w cos )
To eliminate the total energy and the momentum of the electron from our equations, we
make use of the relativistic equation which relates the total energy of the electron to the
momentum of the electron
I/# œ :/# - #  7/# - %
Simply plugging in the expressions for the total energy of the electron and the
momentum of the electron, we obtain
:o  :w -  7/ - # ‘ œ ’:o#  :w  #:o :w cos )“- #  7#/ - %
#
#
Expanding the left-hand side of this equation, cancelling 7#/ - % on both sides, and
dividing by c# gives
:o  : w
Expanding :o  :w
#
#
#
 # :o  :w 7/ - œ :o#  :w  #:o :w cos )
and regrouping gives
:o :w "  cos )  :o  :w 7/ - œ !
Now we substitute the relation : œ 2Î- for the momentum of the photons and obtain
2 2
2
2
"  cos ) œ Œ  w 7/ w
-o -o
which can be simplified to the form of Compton's equation
- w  -o œ
2
"  cos )
7/ -
Now the combination of constants 2Î7/ - must have units of length. By multiplying the
top and the bottom of this group by - we obtain
2"#%! /Z  87
œ
œ #Þ%$ ‚ "!$ 87
7/ - #
&"" ‚ "!$ /Z
Evidence for Photons
18
This is the so-called G97:>98 A+@/6/81>2 90 >2/ /6/-><98. Notice that the maximum
change in wavelength occurs for backscattered electrons, and the maximum change in
wavelength is only twice the Compton wavelength of the electron - which is very small.
The Compton scattering equation can also be written in terms of the incident and
scattered photon energies. By doing this, we can determine the energy given to the target
electron in the collision process. The maximum energy imparted to the electron occurs
when the photon is scattered directly backward (at ) œ ")!‰ Ñ. This means that the
maximum energy that can be imparted to an electron in the target material has a simple
dependence upon the energy of the incident photon. The maximum energy which can be
imparted to an electron within a target material shows up in gamma-ray spectra as the
“Compton knee” for a given photon.
Exercise: Write out an expression for Compton scattering in terms of the
energies of the incident and scattered photons. Using this expression
determine the maximum energy that can be imparted to a target electron as a
function of the incident photon.