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# Nuclear Spin

Many atomic nuclei have a property called spin: the nuclei behaves as if they were spinning. In fact, any atomic nucleus that possesses either odd mass, odd atomic number, or both has a quantized spin angular momentum and a magnetic moment. The important consequence of nuclear spin is the energy splitting of degenerate Nuclear Spin states under an external magnetic field. It is possible to drive transition between different spin states by applying radio-frequency electromagnetic radiation. In this section we will learn more about nuclear spin.

## What is Nuclear Spin?

The Nuclear Spin is different from the electron spin. The nuclear spin represents the total angular momentum of the nucleus. It is represented by symbol, $I$. The nucleus is, although, composed of neutrons and protons but it acts as if it is a single entity which has intrinsic angular momentum.

The nuclear spin depends on the mass number, if the mass number is odd then the nucleus has half-integer spin like the electron while if the nucleus has even mass number then its spin will be integer spin.

## Nuclear Spin States

The nuclear spin states can take any number, fraction or integer. The number is dependent on the three points,
1. If both the neutrons and the protons in the nucleus are even in number then the nucleus has NO spin states.
2. If the sum of the neutrons and protons in the nucleus is odd then the nucleus has half integer spin (1/2. 3/2, 5/2, …)
3. If both the neutrons and the protons in the nucleus are odd in number then the nucleus has an integer spin states (1, 2, 3, …)
In other words the nucleus with odd number of protons or neutron or both should have the nuclear spin while if both are even then there is no nuclear spin.

## Nuclear Spin Table

The nuclear spin of the various element of the periodic table is shown below. As shown the elements in the periodic tables are divided in three categories. The color code and their corresponding allowed nuclear spin is shown in the figure. There are certain elements for which the value of $I$ is unknown.

## Quantum Number

Quantum Numbers are a set of values which describes the state of any electron including the orientation and type of orbital where it is likely to be found, its spin and its distance from the nucleus.

## Nuclear Spin Quantum Number

The nuclear spin quantum number $l$, determines the allowed spin states of the nucleus and it is represented by; Allowed Spin States = $2l + 1$,

In the case of hydrogen, $l$ = $\frac{1}{2}$ and hence the allowed spin states of the nucleus is 2. Similarly if the value of $l$ of an element is known we can find the allowed states of the nucleus of that atom.

## Nuclear Spin Angular Momentum

The nuclear angular momentum is the angular momentum which is possessed by some nuclei. The nuclear spin angular momentum has two quantum number associated with it, a spin quantum number $I$ and magnetic moment quantum number $m_{I}$. The magnitude of the nuclear spin angular momentum is given by;

Magnitude of Nuclear Spin Angular Momentum = $ħ\sqrt{I(I+1)}$

and the z – component of the nuclear spin angular momentum is given by;

z – component of the nuclear spin angular momentum = $m_{I}ħ$

The value of $m_{I}$ is given by $-I$ ≤ $m_{I}$ ≤ $+I$. From this equation

The number of spin of a nucleus is = $2I + 1$;

As we know that the nucleus has protons and neutrons and since protons and neutrons has $I$ = $\frac{1}{2}$, so the allowed states of the numbers of spin of nucleus are 2.

As we know that the nuclei is the assembly of the protons and neutron. The entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to their angular momentum which usually is denoted by $I$. If the total angular momentum of a neutron is jn = ℓ + s and for a proton is jp = ℓ + s (where s for protons and neutrons happens to be $\frac{1}{2}$ same as that of electron) then the nuclear angular momentum quantum numbers $I$ are given by:

$I$ = |jn - jp|, |jn - jp| + 1, |jn - jp| + 2,........, |jn - jp| - 2, |jn - jp| - 1, |jn - jp|

The nuclear spin angular momentum for the hydrogen is:

1H1 $I$ = $(\frac{1}{2})^{+}$
1H2 $I$ = $1^{+}$
1H3 $I$ = $(\frac{1}{2})^{+}$

## Nuclear Spin Calculations

There are several methods for finding the nuclear spin of the given atom. The Stern-Gerlach experiment may be used to calculate the spin of the nucleus. This experiment is used to find the electron and atomic spin of the elements. The experiment was named after Otto Stern and Walther Gerlach and it is conducted by them in 1922. This was the first attempt to show that the elementary particles possess the quantum properties.

The Stern-Gerlach experimental apparatus consists essentially of a magnet which produces a non uniform magnetic field. A beam of atoms enters from the one side of the magnet in a direction perpendicular to the magnetic field. As a consequence of the interaction of nuclear and electronic spin with the applied magnetic field, the atoms undergo the deflection, as they go through the field. As they come out of the other side of the magnet, the atoms are detected by counters, possibly acting also as filters.

The particles (or atoms) spin is calculated by checking whether the particle (or atom) is deflected in the non uniform magnetic field. If the magnetic moment of the particle (or atom) is aligned with that of magnetic field than there is no deflection else it will deflect depending upon the net magnetic field on the particle (or atom). The deflection may be upwards or downwards.

This experiment showed that the electron (or atom) has some total angular momentum while earlier it was understood that the electron (or atom) has zero total angular momentum. If that was the case then the particles should not deflect when they are subjected to the magnetic field.