Quantum Interpretation
3.8 Compton Effect
When a photon enters matter, it is likely to interact with one of the atomic elec-trons. According to classical theory, the electrons will oscillate at the photon frequency because of the interaction of the electron with the electric and mag-netic field of the photon and will reradiate electromagmag-netic radiation (photons) at this same frequency. This is called Thomson scattering. However, in the early 1920s Arthur Compton experimentally confirmed an earlier observation by J. A.
Gray that, especially at backward-scattering angles, there appeared to be a com-ponent of the emitted radiation (called a modified wave) that had a longer wavelength than the original primary (unmodified) wave. Classical electromag-netic theory cannot explain this modified wave. Compton then attempted to understand theoretically such a process and could find only one explanation:
Einstein’s photon particle concept must be correct. The scattering process is shown in Figure 3.20.
Compton proposed in 1923 that the photon is scattered from only one elec-tron, rather than from all the electrons in the material, and that the laws of the conservation of energy and momentum apply as in any elastic collision between two particles. We recall from Chapter 2 that the momentum of a particle moving at the speed of light (photon) is given by
p ! E c !hf
c ! h
l (3.38)
Thomson scattering Explain how the Duane-Hunt rule can be used to determine
the electron bombarding energy in a device such as a scan-ning electron microscope.
Solution If we look closely at Equation (3.37), we can see that any reduction in the acceleration voltage V0 will lead to an increase in the value of lmin. A careful analysis of the
minimum value of the wavelength should be in agreement with the expected voltage V0. If the value of lmin varies over time, for example depending on the electron beam current, it may be due to anomalous charging effects in the beam acceleration/transport system. Solutions for problems like this require painstaking efforts and may dictate the experi-mental conditions, such as using lower beam currents to avoid problems.
C O N C E P T U A L E X A M P L E 3 . 1 4
If we have a tungsten anode (work function f ! 4.63 eV) and electron acceleration voltage of 35 kV, why do we ig-nore in Equation (3.36) the initial kinetic energy of the electrons from the filament and the work functions of the filaments and anodes? What is the minimum wavelength of the x rays?
Strategy We can ignore the initial electron kinetic ener-gies and the work functions, because they are on the order of a few electron volts (eV), whereas the kinetic energy of the electrons due to the accelerating voltage is 35,000 eV.
The error in neglecting everything but eV0 is small. We will use Equation (3.37) to determine the minimum wavelength.
Solution We use the Duane-Hunt rule of Equation (3.37) to determine
lmin!1.240 $ 10#6 V
#
m35.0 $ 103 V !3.54 $ 10#11 m which is in good agreement with the data of Figure 3.19.
E X A M P L E 3 . 1 5
We treat the photon as a particle with a definite energy and momentum. Scat-tering takes place in a plane, which we take to be the xy plane in Figure 3.20.
Both x and y components of momentum must be conserved, because of the vec-tor nature of the linear momentum. The energy and momentum before and after the collision (treated relativistically) are given in Table 3.4. The incident and scattered photons have frequencies f and f *, respectively. The recoil elec-tron has energy Ee and momentum pe.
In the final system the electron’s total energy is related to its momentum by Equation (2.70):
Ee2!1mc222"pe2c2 (3.39) We can write the conservation laws now, initial = fi nal, as
Energy hf " mc2!hfœ"Ee (3.40a) the photon. We first eliminate the recoil angle f by squaring Equations (3.40b) and (3.40c) and adding them, resulting in
pe2! ah scatter-ing of a photon by an electron es-sentially at rest.
Arthur Compton (1892– 1962) is shown here in 1931 looking into an ionization chamber that he de-signed to study cosmic rays in the atmosphere. Compton re-ceived his degrees from the Col-lege of Wooster and Princeton University. He spent most of his career at the University of Chi-cago and Washington University, St. Louis. After his early work with x rays for which he received the Nobel Prize in 1927, he was a pioneer in high-energy physics through his cosmic ray studies.
Compton was also a leader in the establishment of the Manhattan Project to produce
the atomic bomb during World War II and, afterwards, for nuclear power generation.
Energy or Momentum Initial System Final System
Photon energy hf hf *
Then we substitute Ee from Equation (3.40a) and pe from Equation (3.41) into Equation (3.39) (setting l ! c/f ).
3h1f # fœ2 " mc242!m2c4" 1hf22"1hfœ22#21hf2 1hfœ2cos u Squaring the left-hand side and canceling terms leaves
mc21f # f œ2 ! hffœ11 # cos u2 Rearranging terms gives
h
mc2 11 # cos u2 !f # f œ ff œ !
c l# c
lœ c2 llœ
! 1
c1lœ# l2
or
¢l ! lœ# l ! h
mc11 # cos u2 (3.42)
which is the result Compton found in 1923 for the increase in wavelength of the scattered photon.
Compton then proceeded to check the validity of his theoretical result by performing a careful experiment in which he scattered x rays of wavelength 0.071 nm from carbon at several angles. He showed that the modified wave-length was in good agreement with his prediction.* A part of his data is shown in Figure 3.21, where both the modified (lœ) and unmodified (l) scattered waves are identified.
Compton effect
*An interesting personal account of Compton’s discovery can be found in A. H. Compton, American Journal of Physics 29, 817– 820 (1961).
Figure 3.21 Compton’s origi-nal data showing (a) the primary x-ray beam from Mo unscattered and (b) the scattered spectrum from carbon at 135° showing both the modified and unmodified wave. Adapted from Arthur H. Comp-ton, Physical Review 22, 409-413 (1923).
Molybdenum Ka!
line, primary
Modified
Scattered!
from carbon!
at 135°!
Unmodified
Glancing angle from calcite!
(proportional to wavelength)
6°30* 7°! 7°30*
Intensity
(b) (a)
The kinetic energy and scattering angle of the recoiling electron can also be calculated. Experiments in which the recoiling electrons were detected were soon carried out, thus completely confirming Compton’s theory. The process of elastic photon scattering from electrons is now called the Compton effect. Note that the difference in wavelength, ¢l ! lœ# l, depends only on the constants h, c, and me in addition to the scattering angle u. The quantity lC ! h/mec ! 2.426
$ 10#3 nm is called the Compton wavelength of the electron. Only for wave-lengths on the same order as lC (or shorter) will the fractional shift %l/l be large. For visible light, for example with l ! 500 nm, the maximum %l/l is on the order of 10#5 and %l would be difficult to detect. The probability of the oc-currence of the Compton effect for visible light is also quite small. However, for x rays of wavelength 0.071 nm used by Compton, the ratio of %l/l is %0.03 and could easily be observed. Thus, the Compton effect is important only for x rays or g-ray photons and is small for visible light.
The physical process of the Compton effect can be described as follows. The photon elastically scatters from an essentially free electron in the material. (The photon’s energy is so much larger than the binding energy of the almost free electron that the atomic binding energy can be neglected.) The newly created scattered photon then has a modified, longer wavelength. What happens if the photon scatters from one of the tightly bound inner electrons? Then the binding energy is not negligible, and the electron might not be dislodged. The scattering in this case is effectively from the entire atom (nucleus " electrons). Then the mass in Equation (3.42) is several thousand times larger than me, and %l is cor-respondingly smaller. Scattering from tightly bound electrons results in the unmodified photon scattering (l! lœ), which is also observed in Figure 3.21.
Thus, the quantum picture also explains the existence of the unmodified wave-length predicted by the classical theory (Thomson scattering) alluded to earlier.
The success of the Compton theory convincingly demonstrated the cor-rectness of both the quantum concept and the particle nature of the photon.
The use of the laws of the conservation of energy and momentum applied rela-tivistically to pointlike scattering of the photon from the electron finally con-vinced the great majority of scientists of the validity of the new modern physics.
Compton received the Nobel Prize in Physics for this discovery in 1927.
Compton wavelength
An x ray of wavelength 0.050 nm scatters from a gold target.
(a) Can the x ray be Compton-scattered from an electron bound by as much as 62 keV ? (b) What is the largest wave-length of scattered photon that can be observed? (c) What is the kinetic energy of the most energetic recoil electron and at what angle does it occur?
Strategy We first determine the x-ray energy to see if it has enough energy to dislodge the electron. We use Equa-tion (3.42) with both the atomic and electron mass to determine the scattered photon wavelength. We then use the conservation of energy to determine the recoil electron kinetic energy.
Solution From Equation (3.35) the x-ray energy is Ex ray!1.240 $ 103 eV
#
nm0.050 nm !24,800 eV ! 24.8 keV Therefore, the x ray does not have enough energy to dis-lodge the inner electron, which is bound by 62 keV. In this case we have to use the atomic mass in Equation (3.42), which results in little change in the wavelength (Thomson scattering).
Scattering may still occur from outer electrons, so we examine Equation (3.42) with the electron mass. The lon-gest wavelength lœ! l " %l occurs when %l is a maximum or when u ! 180°.