Protons and neutrons (together called nucleons) are held...

Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons - hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is $$U=U_{0}\left[1-e^{-x / x_{0}}\right]$$ where $x$ is the distance between the centers of the two nucleons, $x_{0}$ is a constant having the value $x_{0}=2.0 \times 10^{-15} \mathrm{m},$ and $U_{0}=6.0 \times 10^{-11} \mathrm{J}$ a. Calculate and draw an accurate potential-energy curve from $x=0 \mathrm{m}$ to $x=10 \times 10^{-15} \mathrm{m} .$ Either calculate about 10 points by hand or use computer software. b. Quantum effects are essential for a proper understanding of how nucleons behave. Nonetheless, let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass $1.67 \times 10^{-27} \mathrm{kg}$ and diameter $1.0 \times 10^{-15} \mathrm{m}$ (We will consider neutrons rather than protons so as to avoid complications from the electric forces between protons.) You are going to hold two neutrons $5.0 \times 10^{-15} \mathrm{m}$ apart, measured between their centers, then release them. Draw the total energy line for this situation on your diagram of part a. c. What is the speed of each neutron as they crash together? Keep in mind that both neutrons are moving.

Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons - hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is
$$U=U_{0}\left[1-e^{-x / x_{0}}\right]$$
where $x$ is the distance between the centers of the two nucleons,
$x_{0}$ is a constant having the value $x_{0}=2.0 \times 10^{-15} \mathrm{m},$ and $U_{0}=6.0 \times 10^{-11} \mathrm{J}$
a. Calculate and draw an accurate potential-energy curve from $x=0 \mathrm{m}$ to $x=10 \times 10^{-15} \mathrm{m} .$ Either calculate about
10 points by hand or use computer software.
b. Quantum effects are essential for a proper understanding of how nucleons behave. Nonetheless, let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass $1.67 \times 10^{-27} \mathrm{kg}$ and diameter $1.0 \times 10^{-15} \mathrm{m}$ (We will consider neutrons rather than protons so as to avoid complications from the electric forces between protons.) You are going to hold two neutrons $5.0 \times 10^{-15} \mathrm{m}$ apart, measured between their centers, then release them. Draw the total energy line for this situation on your diagram of part a.
c. What is the speed of each neutron as they crash together? Keep in mind that both neutrons are moving.

Key Concepts

Kinetic Energy and Momentum

Kinetic energy is a measure of the energy associated with an object’s motion and is typically calculated using the equation (1/2)mv², where m is mass and v is velocity. Momentum, which is conserved in isolated systems, ties into these calculations as each moving particle contributes to the overall energy and dynamics. These concepts are essential when determining the speeds of particles after potential energy is converted to kinetic energy in collision or release events.

Energy Conservation Principle

The energy conservation principle is a fundamental concept in physics stating that the total energy within an isolated system remains constant over time. In collision scenarios involving nucleons or other particles, the decrease in potential energy is converted into kinetic energy. This principle allows the calculation of particle speeds after interaction by equating the energy lost in potential form to the kinetic energy gained.

Potential Energy Curve

A potential energy curve visually represents how the potential energy of a system varies with the distance between interacting particles. In the context of nucleons, the curve illustrates how the strong force changes from a high binding energy at close separations to lower binding effects as the distance increases, providing insight into the equilibrium spacing and stability of the nucleus.

Strong Nuclear Force

The strong nuclear force is the fundamental interaction that binds nucleons (protons and neutrons) together within the nucleus. It overcomes the repulsive electromagnetic force between protons at very short distances, making it crucial for the stability of atomic nuclei. While it is very powerful at short ranges, it rapidly weakens as nucleons move apart.

Exponential Potential Energy Model

The exponential potential energy model is commonly used to describe the interaction between nucleons under the strong force. This model expresses the potential energy as a function of distance with an exponential decay term, capturing the rapid decrease in force strength as the separation between the particles increases. It provides a mathematical framework for predicting interactions and energy states in nuclear physics.

Transcript

00:01 In this exercise, we need to remember that the potential energy u of x is an energy associated with a force, such that the force of x is equal to minus the derivative of the potential energy u.

00:22 And we can write the conservation of energy as the initial kinetic energy, k -0 plus the initial potential energy u -0 is equal to the final kinetic energy kf plus the final potential energy uf.

00:40 And in our problem, we have, we model actually the potential between two nucleants to be equal to zero times one minus z to the minus x over x zero.

00:58 This is the potential generated by the strong force between the two nucleons where u0 is equal to 6 times cents of the minus 11 joules x0 is equal to 2 times center to minus 15 meters this is a typical nuclear distance and we also have the information that the mass of a nucleon is 1.

01:36 0 .67 times 10 to the minus 27 kilograms.

01:44 And also we have to model them as having a diameter of 1 times 10 to the minus 15 meters.

01:56 And in question a, our goal is to draw the potential energy, this potential energy here, either by hand or by using a software.

02:09 I chose to use a computer program.

02:12 You could do it by hand.

02:15 The situation would be the same.

02:18 I just plotted it using python, but you could do it using mathematica, matlab, or whatever program you are most familiar with.

02:32 So this here is our plot.

02:35 Notice that the potential energy always increases with x.

02:42 But it tends to a plateau.

02:45 So it will never be greater than 6 times 10 to the minus 11 joules.

02:52 Then in question b, we have to suppose that you hold the two nucleants at a distance of 5 times 10 to minus 11 meters apart, counting from their centers.

03:18 And we have to calculate the potential energy u and draw it in the graph.

03:25 So the potential energy u is u 0, 6 times 10 to minus 11 times 1 minus the exponential of 5 times 10 to the minus 11 divided by 2 times 10 to the minus 15 and actually i'm sorry this shouldn't be 5 2 times 10 to the minus 11.

03:50 Should be 5 times 0 to the minus 15 meters...

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