A photon of wavelength 0.14800 nm, traveling due east, is scattered by an electron initially at

A photon of wavelength 0.14800 nm, traveling due east, is...

Key Concepts

Compton Scattering

This concept refers to the scattering of a photon by a charged particle, typically an electron, where the photon loses energy and its wavelength increases. The process is governed by the Compton formula, which relates the change in wavelength to the scattering angle and fundamental constants such as Planck’s constant and the electron mass. It illustrates the particle-like behavior of light and is a key experimental confirmation of quantum mechanics.

Photon Momentum

Photon momentum is given by the relation p = h/?, where h is Planck’s constant and ? is the wavelength of the photon. Despite having no mass, photons carry momentum, and when they interact with particles like electrons, this momentum is transferred, causing changes in both the photon's path and the motion of the electron. Understanding photon momentum is essential for analyzing energy and momentum conservation in scattering events.

Conservation of Momentum

In scattering problems, the law of conservation of momentum asserts that the total momentum before the interaction equals the total momentum after the interaction. This principle is applied vectorially, meaning both the magnitude and direction of momentum components are conserved. In problems involving photon-electron scattering, momentum conservation helps determine the directions and magnitudes of the particles’ momenta after the event.

Vector Decomposition in Scattering

Vector decomposition involves breaking down momentum vectors into components along chosen coordinate axes. In scattering scenarios where the particles travel in directions not aligned with the coordinate axes, this technique is critical. It allows one to separately analyze the conservation of momentum in the horizontal and vertical directions, which is essential in determining quantities such as scattering angles and the component of the electron's momentum in a specified direction.

Transcript

00:01 For this exercise, we have the system shown here on the figure, where a photon represented by this blue curvy line is incident on an electron.

00:12 And then the photon scatters off at a certain angle theta from the original path.

00:19 And we know the other exercise tells us that the incident photon had a wavelength of 0 .1 .1 .2.

00:32 148 nanometers, which is the same as 1 .48 times 10 to the minus 10 meters.

00:43 And the scattered photon has a wavelength of 1 .149 nanometers.

00:55 And this is 1 .49 times 10 to the minus 10 meters.

01:01 And the first thing we're asked is to calculate the angle theta.

01:07 So according to compton's, the compton's formula we have that the variation in the lambda in the wavelength is given by h over the mass of the electron times the speed of light, one minus the cosine of the scattering angle.

01:31 And we know the variation of the of the wavelength to be 10 to the minus 12 meters.

01:43 That's, uh, you just have to subtract the scattered angle from the incident angle.

01:48 That's going to be 0 .01 times 10 to the minus 10th.

01:53 And that's 10 to the minus 12 meters.

01:57 This is going to be equal to h over mec.

02:01 That constant is just 2 .43 times 10 to the minus 12.

02:08 It's useful to remind ourselves of this, to always have this constant at hand when working with components scattering, times one minus the cosine of theta.

02:25 The 10 to the minus 12th will cancel out here.

02:28 So we're going to have 1 over 2 .43 equals to 1 minus cosine of theta.

02:38 So we're going to have that the cosine of theta is 1 minus 1 over 2 .43.

02:51 And this is 0 .59...

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