Figure 40.17 showed that a typical nuclear radius is 4.0 fm. As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is E_n = −20 MeV. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

Verified step by step guidance

1. Start by understanding the concept of quantum tunneling. The neutron is bound inside the nuclear potential well, and its energy is less than the potential barrier outside the nucleus. This means the neutron can penetrate into the classically forbidden region due to quantum mechanical effects.
2. The penetration distance can be estimated using the wavefunction decay in the forbidden region. The wavefunction decreases exponentially as \( \psi(x) \propto e^{-\kappa x} \), where \( \kappa \) is the decay constant. \( \kappa \) is given by \( \kappa = \sqrt{\frac{2m(U - E)}{\hbar^2}} \), where \( m \) is the mass of the neutron, \( U \) is the potential barrier height, and \( E \) is the neutron's energy.
3. Substitute the known values into the formula for \( \kappa \): \( m \) is the neutron mass (\( 1.675 \times 10^{-27} \ \text{kg} \)), \( U \) is the potential barrier height (assume a typical nuclear potential of \( 40 \ \text{MeV} \)), and \( E \) is the neutron's energy (\( -20 \ \text{MeV} \)). Convert \( U \) and \( E \) to joules using \( 1 \ \text{MeV} = 1.602 \times 10^{-13} \ \text{J} \).
4. Once \( \kappa \) is calculated, the penetration distance \( x \) can be estimated as the distance where the wavefunction has significantly decayed. A common approximation is to use \( x \approx \frac{1}{\kappa} \).
5. Finally, calculate the penetration distance as a fraction of the nuclear radius. Divide the penetration distance \( x \) by the nuclear radius (\( 4.0 \ \text{fm} \), where \( 1 \ \text{fm} = 10^{-15} \ \text{m} \)). This gives the fraction of the nuclear radius that the neutron penetrates into the forbidden region.

Key Concepts

Nuclear Radius

The nuclear radius is a measure of the size of an atomic nucleus, typically on the order of femtometers (fm). It is defined as the distance from the center of the nucleus to the point where the density of nuclear matter drops significantly. Understanding the nuclear radius is crucial for analyzing the spatial extent of nuclear forces and the behavior of particles like neutrons within the nucleus.

Quantum Tunneling

Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential energy barrier that it classically should not be able to surmount. In the context of a neutron in a nucleus, this concept helps explain how neutrons can penetrate into the classically forbidden region outside the nucleus, despite having insufficient energy to escape according to classical physics.

Classically Forbidden Region

The classically forbidden region refers to areas where a particle's energy is less than the potential energy, making it classically impossible for the particle to exist there. In nuclear physics, this region is significant for understanding the behavior of particles like neutrons, as it relates to their probability of being found outside the nucleus and the extent of their penetration into this region.