During a Compton scattering experiment, an electron that was initially at rest recoils in the di

During a Compton scattering experiment, an electron that was...

Key Concepts

Photon Energy-Wavelength Relationship

This concept describes how the energy of a photon is inversely proportional to its wavelength, with the relationship given by E = hc/?. It forms the basis for converting between photon energy and wavelength, an essential step in analyzing phenomena like Compton scattering.

Compton Scattering

Compton scattering is the process in which a photon collides with a particle, typically an electron, resulting in a transfer of energy and momentum. This phenomenon demonstrates the particle-like behavior of light and is a key experimental verification of quantum mechanics.

Compton Wavelength Shift Formula

The Compton wavelength shift formula, ?? = (h/m_e c)(1 - cos?), quantifies the increase in the wavelength of a photon after scattering off an electron. This expression is central to understanding how the scattering angle influences the energy transferred from the photon to the electron.

Conservation of Energy and Momentum in Scattering

In photon-electron collision processes such as Compton scattering, the conservation laws of energy and momentum are fundamental. They govern the redistribution of energy between the photon and the recoiling electron, enabling the calculation of variables like the initial photon wavelength and the kinetic energy acquired by the electron.

Transcript

00:02 In this exercise, we have a photon scattering of an electron, and we have the information that the final electron has a velocity that is in the same direction as the photon's initial momentum.

00:13 We also have the information that the final kinetic energy of the photon, of the electron, i'm sorry, 0 .2 kiloelectron volts.

00:22 And from this, from these information, we can get the fact that the scatterers, photon is going to have to be pointing, the scattered photon's momentum is going to have to be pointing at a direction of 180 degrees from the original photon's path, and that is due to conservation of momentum.

00:49 Since the incident photon and the final electron both point in the same direction, conservation of momentum requires that the scattered photon doesn't have any component in the wide direction because that would make that would not conserve momentum in the wide direction so we so the scattered photon has to have and to make a 180 degrees with the original path so we're going to keep this in mind when making our calculations and the first thing we're going to have to uh to calculate the two things we're going to have to calculate here are lambda, which is the wavelength of the incident photon, and lambda s, which is the wavelength of the scattered photon.

01:47 Now, in order to do that, we're going to make you use first.

01:52 So first, i'm going to write here, compton's formula, which is lambda s equals lambda, plus h over mc, where m is the mass of the electron.

02:03 1 minus cosine of theta.

02:08 And since theta is 180 degrees, the cosine of theta is minus 1.

02:12 So lambda s equals lambda plus 2h over mc.

02:21 Also, so this is the first part, i'm going to highlight it here because this is going to be important.

02:32 In the second part, we have the kinetic energy of the electron, k.

02:37 And we know that from conservation of energy the energy of the incident photon equals the energy of the scattered photon plus the kinetic energy of the electron so that the kinetic energy of the electron is going to be equal to the the energy of the incident photon which is hc over lambda minus the energy of the scarred photon, which is hc over lambda s.

03:06 And i'm going to just go ahead and write it down as hc 1 minus 1 over lambda minus.

03:18 So this is the second thing that you're going to need.

03:22 I'm going to also highlight this.

03:28 And we're going to combine these two equations in order to find lombda.

03:33 Okay.

03:37 So i'm going to write k as hc1 over lambda.

03:45 And then i'm going to write lambda as a function of lambda according to the comptons formula.

03:52 So this is going to be lambda plus 2h over mc.

04:01 Okay.

04:02 And i can write this now as k equals h.

04:12 I'm going to multiply and divide the first term by lambda plus 2h over mc.

04:20 And the second term i'm going to multiply and divide by lambda.

04:23 So the denominator, we're going to have lambda times lambda plus 2h mc.

04:33 And up here we're going to have lambda plus 2hmc minus lambda.

04:44 So this is just going to be k equals.

04:47 2h squared over m the c's will cancel out 1 over lambda squared plus 2h lambda over m c okay so if we rearrange this equation here we're gonna get 2h 2h squared okay i'm gonna multiply both sides by the denominator here, this denominator.

05:29 So we're going to have 2h squared equals m k lambda squared plus 2h over c.

05:46 Vms will cancel out.

05:48 K lambda...

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