تأثير كومبتون Compton-Effect
Arthur H. Compton observed the scattering of x-raysfrom electrons in a carbon target and found scattered x-rays with a longer wavelength than those incident upon the target. The shift of the wavelength increased with scattering angle according to the Compton formula:
Compton explained and modeled the data by assuming a particle (photon) nature for light and applying conservation of energy and conservation of momentum to the collision between the photon and the electron. The scattered photon has lower energy and therefore a longer wavelength according to the Planck relationship.
At a time (early 1920"s) when the particle (photon) nature of light suggested by thephotoelectric effect was still being debated, the Compton experiment gave clear and independent evidence of particle-like behavior. Compton was awarded the Nobel Prize in 1927 for the "discovery of the effect named after him".
Compton Scattering Experiment.
Theoretical Analysis of Compton"s Experimental Observations
Compton used the Einstein’s concept of photon for the first time in the analysis of the observations in his experiments. A photon is a localized packet of energy. It may be considered as a particle of energy E0 given by Planck"s quantum hypothesis E0= hν and momentum p0 given by De Broglie relation p0 = h / λ where ν is the frequency and λ is the wavelength of the X ray photon.
The wavelength of the X ray photons is ≤1A0 and energy hν≥ 104 eV. Due to the high energy of the photons, the velocity acquired by the electrons is very high and comparable to the speed of light. Hence it is necessary to use relativistic expressions for the kinetic energy and momentum of the electrons.
The general equation for the total relativistic energy E in terms of rest mass m0 is given by
Velocity of the photon is c, its energy E0 = hν is finite. Hence the rest mass of photon must be zero. Its energy is entirely kinetic, Momentum p0 of the photon is obtained from the general equation
Momentum of photon p0 is given by p0 = h / λ
The interaction between incident X-ray photon and an electron in the target may be considered as a collision between two particles. Compared to the energy of the X-ray photons, the binding energy of the electrons in the target material is small and they may be treated as free electrons. X-ray Photon having total relativistic energy E0 and momentum p0 strikes a stationary electron having mass me and rest mass energy mec2. The incoming photon gives part of its energy to one of the loosely bound (almost-free electrons) in the outer shell of the atom or molecule in the stationary target This energy of photon is used to release and give kinetic energy to the electrons. After the collision the ejected electron recoils with kinetic energy T2 and momentum p2 in the direction making an angle ф. Its total energy E2is the sum of kinetic energy T2 and rest mass energy mec2. X-ray photon is scattered with the remaining total relativistic energy E1 and momentum p1in a direction making an angle θ with the original direction, This interaction is according to the conservation of linear momentum and total energy in inelastic scattering. As some energy of the scattered X- ray photons is lost (to the recoiling electron), the frequency ν of the scattered X-ray photon is decreased to ν’or its wavelength λ is increased to λ’. It is calculated using the relations as follows
According to the conservation of momentum
Φ is eliminated by squaring and adding above two equations
According to the conservation of total relativistic energy
E0 + mec2 = E1 + T2 + mec2
E0 − E1 = T2
hv − hv" = T2
c(p0 − p1) = T2
Energy of the electron after collision is given by
E2 = T2 + mec2
E2 is also expressed by
where the general equation E2 = c2p2 + (m0c2)2 is applied to the electron
Equating the above two expressions for energy of the electron
mec(p0 − p1) = (p0p1)(1 − cosθ)
Using algebraic relations to eliminate variables Compton arrived at the following relationship between the shift in wavelength and the scattering angle θ in terms of constant parameters as follows
is the initial wavelength of the X-ray photon
is the wavelength after scattering of the X-ray photon
is the Compton wavelength
h is the Planck constant
me is the mass of the electron
c is the speed of light
θ is the scattering angle of the X-ray photon.
It is known as the Compton equation. From the formula, theoretical value of the wavelength for scattering at θ = 900 comes out to be 0.0733 nm. It is consistent with the experimental value 0.0731 nm (refer to the table)
The quantity is called as the Compton wavelength of the electron. Its value is equal to 2.43×10 − 12 m.
The wavelength shift is called as the Compton Shift.
The Compton Shift varies with the scattering angle of photon as follows
It has Minimum value Δλ = zero (for θ = 0°)
when the incident photon is not deflected from its path
It is called as a “Grazing collision” of photon with electron
It has Maximum value = Twice the Compton wavelength (for θ= 180°).
When the incident photon reverses its direction
It is called as a “Head-on collision” of photon with electron
From the graphs it is seen that in addition to the shifted wavelength an unmodified wavelength also is present. It is due to the scattering of X-ray photon from tightly bound electrons. In that case, not a single loose or free electron but the entire atom is involved in the interaction with photon and it recoils. In the formula for mass of atom MA (instead of mass of electronme) is to be used. MA > > me. Hence, is a minute, almost negligible quantity i.e. wavelength remains unaffected after scattering.
Similar reasoning applies if light photons in the visible range are used. They have longer wavelengths of the order of 500 nm. Their energies are far too smaller than X-rays and not sufficient to overcome the binding energies of even the loosely bound electrons. The wavelength shift is very very small in comparison to their wavelength. By using them it is difficult to detect minute wavelength shift .On the contrary, X-rays have short wavelengths of the order of 0.1 nm. Their use makes the measurement of minute wavelength shift (of less than 0.1 nm) comparatively easy. Hence X ray photons were found to be more appropriate to demonstrate the scattering between photons and electrons and confirm the particle nature of light.
1.4.3 The Compton Effect
Compton scattering is a type of inelastic scattering that high energy X-rays, gamma rays undergo in matter. When a X ray or gamma ray photon collides with an electron in matter, part of its energy is transferred to the electron, which recoils and is ejected from its atom (atom becomes ionized). X ray or gamma ray is scattered with the remaining energy. Due to the "degradation" or a decrease in energy, its wavelength is increased. It is called as the Compton effect.
The relationship between the shift in wavelength and the scattering angle θ, called as the Compton shift is given by
The Compton shift depends upon the angle of scattering θ,. Wavelength λ’ of the scattered X ray photon depends upon the angle of scattering θ and also the mass of the recoiling electron