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Nuclear Spin

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Nuclei have an intrinsic angular momentum (spin). This fact leads to important symmetry consequences for some molecules: the nuclear spin wave function of a molecule is included in the total wave function and the total wave function must obey symmetry rules, for example, the Pauli's exclusion principle. The other important consequence is that, the splitting energy of degenerate nuclear spin states under an external magnetic field. The local field experienced by the nuclei, hence the resonant frequencies, depend on details of the molecular structure.

Definition

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Groups of particles also have a spin as a result of the spin of their individual particles. A nucleus, for example, has a spin determined by the spin of the nucleons that make it up. In fact, the nuclear spin is the sum of the orbital angular momentum and the intrinsic spin of the nucleons. For even mass number nuclei, the spin is an integer; for odd mass number nuclei, the spin is a half integer ($\frac{1}{2}$, $\frac{3}{2}$,....). All nuclei with an even number of protons and neutrons have zero spin in their ground state.

States

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Most atomic nuclei possess spin. The nuclear spin quantum number is conventionally denoted as I. The nucleus of the main isotope of hydrogen, 1H, contains a single proton and has I = $\frac{1}{2}$. The spin of other nuclei are formed by combining together the spins of the protons and the neutrons according to the usual rule. Consider, for example, the 2H nucleus, which contains one proton and one neutron. The proton and neutron spins may be combined in a parallel configuration, leading to a nuclear spin I = 1 or in an anti parallel configuration, leading to a nuclear spin I = 0.

Nuclear Spin States

The two nuclear spin states of hydrogen have a large energy difference of ∼$10^{11}$ kJ$mol^{-1}$. This greatly exceeds the energies available to ordinary chemical reactions or usual electromagnetic fields. The nuclear excited states may, therefore, be ignored, except in exotic circumstances. The value of I in the lowest energy nuclear state is called ground state nuclear spin.For deuterium (2H) the ground state nuclear spin is I = 1.

How to Calculate Nuclear Spin?

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We can calculate the nuclear spin by using its number of protons and the number of neutrons. There is no particular formula to calculate this nuclear spin for an element. According to the distribution of unpaired protons in the energy levels we are calculating the spin. If the number of proton and number of neutrons is even, there is no net spin. So, the spin must be zero in this case. If the number of protons is odd, or it is unpaired, the spin value should be $\frac{1}{2}$, $\frac{3}{2}$,...according to the element.