Question

Isospin, or isotopic spin, is a quantum variable describing the relationship between protons and neutrons in nuclear and particle physics. (Strictly, it describes the relationship between up and down quarks, as described in Chapter \(39,\) but it was introduced before the advent of the quark model.) It has the same algebraic properties as quantum angular momentum: A proton and a neutron form an iso-doublet of states, with total isospin quantum number \(\frac{1}{2} ;\) the proton is in the \(t_{z}=+\frac{1}{2}\) state, and the neutron is in the \(t_{z}=-\frac{1}{2}\) state, where \(z\) refers to a direction in an abstract isospin space. a) What isospin states can be constructed from two nucleons, that is, two particles with \(t=\frac{1}{2}\) ? To what nuclei do these states correspond? b) What isospin states can be constructed from three nucleons? To what nuclei do these correspond?

Short answer

Answer: The possible isospin states for two-nucleon systems are Helium-2 (He-2), Neutron-2 (n-2), and Deuteron (d). For three-nucleon systems, the possible isospin states are Lithium-3 (Li-3), Helium-3 (He-3), Tritium (T), and Neutron-3 (n-3).

Step-by-step solution

Identify possible nucleons and their isospinos

We are given that a proton has an isospin \(t=1/2\) and a projection \(t_z=+1/2\), while a neutron has an isospin \(t=1/2\) and a projection \(t_z=-1/2\).

Calculate isospin states for two nucleons

We need to find the possible isospin states of two nucleons. To do this, we will use the Clebsch-Gordan coefficients for the combined isospinos: 1. For two protons: \(t_1 = t_2 = 1/2\), \(t_{z1} = +1/2\) and \(t_{z2} = +1/2\). The combined state will be \(T=1\) and \(T_z=+1\), and this corresponds to a nucleus with two protons (Helium-2, He-2) 2. For two neutrons: \(t_1 = t_2 = 1/2\), \(t_{z1} = -1/2\) and \(t_{z2} = -1/2\). The combined state will be \(T=1\) and \(T_z=-1\), and this corresponds to a nucleus with two neutrons (Neutron-2, n-2) 3. For a proton and a neutron with \(t_1 = t_2 = 1/2\), \(t_{z1} = +1/2\), and \(t_{z2} = -1/2\). The possible combined states will be \(T=1\) and \(T_z=0\) (formed by adding the isospins) or \(T=0\) and \(T_z=0\) (formed by subtracting the isospins). This corresponds to a nucleus with one proton and one neutron (Deuteron, d)

Calculate isospin states for three nucleons

We need to find the possible isospin states for three nucleons. We will combine them in the following ways: 1. Three protons: \(T_z=+3/2\), as all protons contribute \(+1/2\). This corresponds to a nucleus with three protons (Lithium-3, Li-3) 2. Two protons and one neutron: \(T_z=+1/2\), as the protons contribute \(+1/2\) each and the neutron contributes \(-1/2\). This corresponds to a nucleus with two protons and one neutron (Helium-3, He-3) 3. One proton and two neutrons: \(T_z=-1/2\), as the protons contribute \(+1/2\) each and the neutrons contribute \(-1/2\) each. This corresponds to a nucleus with one proton and two neutrons (Tritium, T) 4. Three neutrons: \(T_z=-3/2\), as all neutrons contribute \(-1/2\). This corresponds to a nucleus with three neutrons (Neutron-3, n-3)

Conclusion

In this exercise, we have found the possible isospin states for two-nucleon systems (He-2, n-2, and d) and three-nucleon systems (Li-3, He-3, T, and n-3).

Key concepts

Quantum Angular Momentum

When we explore the microcosm of atoms and particles, we enter a world where the usual rules of classical physics don't fully apply. Quantum angular momentum is a fundamental concept in this quantum domain. Unlike the angular momentum we encounter in everyday life, which is derived from mass and rotational velocity, quantum angular momentum is an intrinsic property of particles, like electrons, protons, and neutrons.

Every particle has a quantized amount of angular momentum that can be described by 'quantum numbers'. The most important of these is the spin quantum number, which for fundamental particles is either \(\frac{1}{2}\) or an integer. For instance, electrons, protons, and neutrons are fermions with a spin of \(\frac{1}{2}\). In quantum mechanics, angular momentum is not just an important concept for understanding the spin of particles, but is also crucial for determining how these particles interact and the energy levels they can exist in. This concept becomes particularly interesting when applied to nuclear physics, interfacing with the idea of nuclear isospin states.

Nuclear Isospin States

Isospin is a concept in nuclear physics that extends the idea of quantum angular momentum to describe protons and neutrons. This quantum number was introduced to account for the strong nuclear force, which behaves similarly towards both protons and neutrons despite their charge difference.

Protons and neutrons can be seen as two different states of the same particle, known as nucleons, and these states are characterized by isospin quantum numbers. The third component of isospin, typically denoted as \(t_z\), takes a value of \(+\frac{1}{2}\) for protons and \(−\frac{1}{2}\) for neutrons. The total isospin \(T\) arises from the combination of individual isospins in a nucleus and can be useful in predicting the possible states of a system of nucleons. For example, when combining two nucleons, their isospins can align in parallel (resulting in a triplet with \(T = 1\)) or antiparallel (resulting in a singlet with \(T = 0\)) configurations. These different isospin states can correspond to different nuclides, or types of atomic nuclei, with varying numbers of protons and neutrons.

Clebsch-Gordan Coefficients

The Clebsch-Gordan coefficients arise when combining quantum states with angular momentum, such as those in atomic orbits or, pertinent to our discussion, nuclear isospin states. These coefficients serve as a mathematical tool to calculate the result of coupling two quantum states with their associated angular momenta.

To understand their role in nuclear physics, let's consider a nucleus composed of multiple nucleons. Each nucleon carries its isospin, and when nucleons are combined, their individual isospins add up to form the total isospin of the nucleus. The task, then, is to determine the possible combined states and their probabilities. This is where Clebsch-Gordan coefficients come into play: they provide a rigorous way to combine different isospins and predict the resulting isospin state of the nucleus. They are particularly useful in the calculation of isospin in multi-nucleon systems as shown in the exercise solution for both two-nucleon and three-nucleon combinations. Using these coefficients, physicists can predict the allowed isospin states and their corresponding nuclei, such as He-2, n-2, d, Li-3, He-3, T, and n-3.