The nuclei ^5 B^11 and ^6 C^11 are said to be a pair of mirror nuclei because they have the s

The nuclei ^5 B^11 and ^6 C^11 are said to be a pair of...

The nuclei ${ }^{5} \mathrm{~B}^{11}$ and ${ }^{6} \mathrm{C}^{11}$ are said to be a pair of mirror nuclei because they have the same number of nucleons, and the number of protons in one equals the number of neutrons in the other. If nuclear forces are charge independent, their total binding energies should differ only in that the Coulomb energy is higher in ${ }^{6} \mathrm{C}^{11}$. The atomic mass of ${ }^{5} \mathrm{~B}^{11}$ is $11.009305 u$, and the atomic mass of ${ }^{6} \mathrm{C}^{11}$ is $11.011432 u$. (a) Evaluate the difference in their total binding energies. (b) Assuming both nuclei to be uniformly charged spheres of the same radius $r^{\prime}$, and using the expression for the Coulomb energy given in Problem 15 , find the value of $r^{\prime}$ that leads to a difference in Coulomb energy that agrees with the difference in binding energy. (c) Compare this charge distribution radius with the radial dependence of the charge density for the similar nucleus ${ }^{6} \mathrm{C}^{12}$ shown in Figure 15-6.

Key Concepts

Mirror Nuclei

Mirror nuclei are pairs of nuclei in which the number of protons in one nucleus is equal to the number of neutrons in the other and vice versa. This concept is important because, under the assumption that nuclear forces are charge independent, their binding energies should be nearly identical except for differences arising due to Coulomb effects, allowing us to study the influence of electromagnetic repulsion in nuclear systems.

Binding Energy

Binding energy is the energy required to disassemble a nucleus into its individual nucleons. It reflects the stability of the nucleus and is determined by the mass difference between the nucleus and the sum of the masses of its constituent protons and neutrons. Discrepancies in binding energy between mirror nuclei can provide insights into additional contributions such as Coulomb energy.

Coulomb Energy

Coulomb energy refers to the electrostatic repulsion energy between protons in a nucleus. It is a critical concept when analyzing nuclear stability, especially in mirror nuclei where differences in proton numbers cause variations in Coulomb repulsion. The energy can be estimated using models such as that of a uniformly charged sphere, which relates the charge distribution to the radius of the nucleus.

Uniformly Charged Sphere Model

The uniformly charged sphere model is used to approximate the distribution of protons within a nucleus. In this model, the nucleus is considered as a sphere with a uniform charge density. This allows for the derivation of an expression for the Coulomb energy, which depends on both the total charge and the radius of the sphere, facilitating comparisons with experimental binding energy differences.

Charge Independence of Nuclear Forces

The charge independence of nuclear forces is the principle that the strong nuclear interaction affecting nucleons is nearly the same regardless of whether the nucleons are protons or neutrons. This principle underlies the expectation that mirror nuclei should have similar nuclear binding energies aside from the Coulomb energy contributions arising from different proton numbers.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that the nuclei of borough and carbon are saying to be pair of mirror nuclei because they have the same number of nucleons.

00:13 And the number of protons in one equals the number of neutrons in the other.

00:19 If nuclear forces are charged independent, their total binding energies should differ only in that the column energy is higher, in carbon 611.

00:33 Now the atomic mass of 5 borough 11 is given and that it has a value of 11 .0 .0 .0 .93 .0 units of mass.

00:57 And in the case of 6 carbon 11 .11.

01:04 Its math is 11 .011432 units of mass.

01:15 Now for part a of this problem we are asked to evaluate the difference in their total biting energies.

01:25 So to obtain this we do this following procedure.

01:29 We know that the divided energy for borough is equal to five times the mass of hydrogen plus six times the mass of a neutron and this minus the mass of borough and this times the speed of light to the square.

01:53 Now we just simply substitute all of these values.

02:00 So in here we're going to obtain the following.

02:05 The sum of these two terms in here give us a value of 11 .091, 118.

02:22 And this minus the mass of borough, which is 11 .009, 305.

02:32 And this times the speed of light to the square.

02:34 We know that the speed of light is 931 .5 megalletron bolts.

02:41 So with this, we will have that the energy of borough is equal to 76 .209 mega electron balls...