An X-ray photon scatters from a free electron at rest at an...

An X-ray photon scatters from a free electron at rest at an angle of $115^{\circ}$ relative to the incident direction. (a) If the scattered photon has a wavelength of $0.320 \mathrm{nm},$ what is the wavelength of the incident photon? (b) Determine the energy of the incident and scattered photons. ( c) Find the kinetic energy of the recoil electron.

An X-ray photon scatters from a free electron at rest at an angle of $115^{\circ}$ relative to the incident direction. (a) If the scattered
photon has a wavelength of $0.320 \mathrm{nm},$ what is the wavelength of
the incident photon? (b) Determine the energy of the incident
and scattered photons. ( c) Find the kinetic energy of the recoil
electron.

Key Concepts

Compton Wavelength

The Compton wavelength is a fundamental constant specific to a particle, such as an electron, and represents a scale at which quantum effects become significant in scattering events. It appears in the Compton scattering formula and quantifies the maximum possible change in the wavelength of the photon as a function of the scattering angle.

Conservation of Energy and Momentum

The scattering process entails the conservation of both energy and momentum, meaning the total energy and momentum before and after the interaction are equal. These conservation laws are applied to relate the energy lost by the photon to the kinetic energy acquired by the recoiling electron, making them essential for determining the outcomes in a scattering event.

Photon Energy-Wavelength Relationship

In quantum mechanics, the energy of a photon is inversely proportional to its wavelength, as described by the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This relationship is crucial for converting between the energy and wavelength of photons in scattering problems.

Compton Scattering

This is a quantum mechanical phenomenon where a photon scatters off a target particle, such as a free electron, resulting in a change in the photon's wavelength and a corresponding transfer of energy and momentum. The phenomenon is characterized by the Compton formula, which relates the change in wavelength to the scattering angle and the intrinsic Compton wavelength of the electron.

Transcript

00:01 Well, x -ray photons scatters at 105 degrees relative to the incident direction.

00:13 And the photon scatters from a free electron address.

00:20 So 115 degrees is the scattering angle.

00:28 Part a, wavelength of scattered photon is 0 .3 to 0 .0 .0 .0.

00:35 Nanometers so web length of scattered photon lambda prime equals 0 .320 nanometers and 0 .3 to 0 nanometers is equal to 0 .320 0 multiplied by 10 to the power minus 9 meters.

01:06 Well now let's calculate incident photons web length.

01:11 We will find lambda.

01:15 Lambda is the wavelength of incident photon.

01:20 We know that change in wavelength equals h divided by mass of electron times speed of light into 1 minus cosine theta.

01:33 And in this case, theta equals 115 degrees.

01:38 Therefore, h divided by mass of electron times speed of light into one minus cosine of 115 degrees.

01:52 And this gives us delta lambda change in the valent equals h divided by mass of electron times speed of light into one plus 0 .4.

02:13 And then we have change in vablen equals 1 .4, multiply by 6 .63, multiply by 10 to the power minus 34, divided by mass of electron is 9 .1, multiplied by 10 to the power minus 31.

02:36 And then speed of light is 3 multiplied by 10 to the power 8 meter per second.

02:45 Change in web length is equal to 3 .4, 3 .4 multiplied by 10 to the power minus 12 meters.

03:02 Well, change in web length can be written is a lambda prime minus lambda, and therefore, lambda, which is the wavelin to incident photon equals lambda prime minus change in valent and this gives us lambda equals lambda prime is 0 .0 .3 to 0 .3 to 0 multiplied by 10 to the power minus 9 minus then we have change in length and change in mab length is 3 .4 multiplied by 10 to the power minus 12 meters.

03:54 And therefore, lambda equals 3 .17, 3 .17 multiply by 10 to the power minus 10 meters, minus 10 meters.

04:11 And this can be written is lambda equals 0 .0 .317 nanometers.

04:21 Well, lambda equals 0 .317 nanometers.

04:29 Now, let's solve part b...

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