Elements of Modern Physics
42
or
(1 cos )
h
mc
0
λ − λ =
−
θ
(2.34)
This is Compton’s expression for the shift in the wavelength of the scattered
x-rays. It identifies the Compton wavelength as
c
h
mc
λ =
(2.35)
which depends only on the mass of the scattering particle and has a value of
2.43 × 10
–2
Å for the electron. It has an interesting interpretation, that a
photon with wavelength λ c has an energy h ν = mc
2
, i.e., the rest energy of
the particle.
In the discussion presented here, it is assumed that the target electron is
stationary and free. It is also valid if the electron is weakly bound to the
atom with the binding energy of a few eV which is quite small compared
with the energies of the x-ray photons, which are about 10 keV or greater.
However, it may so happen that the electron remains bound in the same
state to the atom, even after the collision with the photon (this is more likely
to happen if the electrons are strongly bound). In this case, the transfer of
energy and momentum is to the atom as a whole, so that the mass of the
atom must be used in place of the mass of the electron. Therefore, the
corresponding Compton wavelength is much smaller (at least by a factor of
1800) and the resulting Compton shift in the wavelength is negligible. This
explains the unshifted component in the spectrum of the scattered x-rays,
which is called the Thomson component.
Some interesting additional features of Compton effect are:
1. The fact that the shift in the wavelength of the radiation is indeed due to
the scattering of the radiation by the electron was confirmed by observing
the scattered electron (Bothe and Geiger, 1925).
2. The main reason for the spread in the wavelength of the Compton-shifted
x-rays is that the initial electron is in general not stationary but has a
momentum spread even inside the atom. The correction due to the binding
of the electron can be taken into account in terms of its momentum
distribution (see Example 4).
3. Though the sift λ – λ 0 is independent of λ 0 , the intensity of scattering
depends on λ 0 . It actually increases as λ 0 → 0 and hence the effect is
more easily observable for x-rays than for lower frequency radiation (indeed
this is the reason why the sky is blue).
4. The scattering angle φ is given by
42
or
(1 cos )
h
mc
0
λ − λ =
−
θ
(2.34)
This is Compton’s expression for the shift in the wavelength of the scattered
x-rays. It identifies the Compton wavelength as
c
h
mc
λ =
(2.35)
which depends only on the mass of the scattering particle and has a value of
2.43 × 10
–2
Å for the electron. It has an interesting interpretation, that a
photon with wavelength λ c has an energy h ν = mc
2
, i.e., the rest energy of
the particle.
In the discussion presented here, it is assumed that the target electron is
stationary and free. It is also valid if the electron is weakly bound to the
atom with the binding energy of a few eV which is quite small compared
with the energies of the x-ray photons, which are about 10 keV or greater.
However, it may so happen that the electron remains bound in the same
state to the atom, even after the collision with the photon (this is more likely
to happen if the electrons are strongly bound). In this case, the transfer of
energy and momentum is to the atom as a whole, so that the mass of the
atom must be used in place of the mass of the electron. Therefore, the
corresponding Compton wavelength is much smaller (at least by a factor of
1800) and the resulting Compton shift in the wavelength is negligible. This
explains the unshifted component in the spectrum of the scattered x-rays,
which is called the Thomson component.
Some interesting additional features of Compton effect are:
1. The fact that the shift in the wavelength of the radiation is indeed due to
the scattering of the radiation by the electron was confirmed by observing
the scattered electron (Bothe and Geiger, 1925).
2. The main reason for the spread in the wavelength of the Compton-shifted
x-rays is that the initial electron is in general not stationary but has a
momentum spread even inside the atom. The correction due to the binding
of the electron can be taken into account in terms of its momentum
distribution (see Example 4).
3. Though the sift λ – λ 0 is independent of λ 0 , the intensity of scattering
depends on λ 0 . It actually increases as λ 0 → 0 and hence the effect is
more easily observable for x-rays than for lower frequency radiation (indeed
this is the reason why the sky is blue).
4. The scattering angle φ is given by