Assuming the nucleus to diffract like a circular disk, use...

Assuming the nucleus to diffract like a circular disk, use the data shown in Figure 12.3 to find the nuclear radius for ${ }^{12} \mathrm{C}$ and ${ }^{16} \mathrm{O}$. How does changing the electron energy from 360 MeV to 420 MeV affect the deduced radius for ${ }^{16}$ O? (Hint: Use the extreme relativistic approximation from Chapter 2 to relate the electron's energy and momentum to find its de Broglie wavelength.)

Key Concepts

Energy-Momentum Relationship

In the extreme relativistic regime, the relation between an electron's energy and its momentum becomes approximately linear (E ? pc). This relationship simplifies the calculation of the de Broglie wavelength from the measured energy, thereby influencing the interpretation of diffraction data to extract information about nuclear size.

Relativistic Electron Scattering

At high energies, electrons must be treated using relativistic mechanics. The extreme relativistic approximation allows the simplification of calculations where the electron's kinetic energy far exceeds its rest mass energy. This is essential when relating the electron’s energy to its momentum in scattering experiments.

De Broglie Wavelength

This fundamental concept of quantum mechanics expresses the wave-like behavior of particles. The de Broglie wavelength (? = h/p) connects a particle’s momentum with its wavelength, and in scattering experiments, it is crucial for interpreting diffraction patterns and determining spatial dimensions such as nuclear radii.

Nuclear Diffraction

This concept involves understanding that when high-energy electrons interact with a nucleus, the scattering pattern can be interpreted using wave phenomena, specifically diffraction. In this context, the nucleus behaves like a diffracting object, and the pattern gives insights into the size and shape of the nucleus.

Circular Disk Model

This model simplifies the nuclear structure by approximating the nucleus as a circular disk. It allows one to use the mathematical formalism of diffraction (similar to that of light through a circular aperture) to relate features in the scattering data, such as intensity minima and maxima, to the nuclear dimensions.

Transcript

00:01 Hi there, so for this problem, assuming the nucleus to defrat like a circular disk, we need to use the data that is shown to find the nuclear radius for 12 carbon and 16 oxygen.

00:23 How does changing the electron energy from 360 mega electron balls to 420? 420 mega electron balls affect the reduced radius for 16 oxygen.

00:50 So we note that for diffraction by a circular disk, the minimum occurs when the angle theta is equal to the sign of minus 1 of 1 .22 times the wavelength divided by the upper toward that distance d.

01:16 And now we know that for 12 carbon add an energy of 420 meg -electron bowls.

01:31 The angle theta is equal to 51.

01:35 So what we are going to do in here is to solve for the diameter of the disk, which is capital d.

01:42 So solving for that, we will find.

01:44 That d is equal to 1 .22 times the wavelength divided by the side of theta.

01:57 And we just put the values in here.

02:01 Well, we don't know the wavelength, but we know that the wavelength can be expressed as the product between plants constant, the speed of light, divided by the energy.

02:12 So we put that in here, sine of theta times the energy.

02:20 And so we now substitute all of these values.

02:24 So we will have 1 .22 times the product between plant's constant and the speed of light.

02:31 That is 1 ,240 mega electron balls times fentometers.

02:38 And this divided by sine of 51 degrees times the energy, which is for 120 megameter...

Please give Ace some feedback