# Nuclear spin

The nuclear spin is the total angular momentum of an atomic nucleus around its center of gravity . Its influence on the properties of macroscopic matter or processes can usually be neglected, which is why it was only discovered in the late 1920s. However, nuclear spin studies are important for understanding the structure of atomic nuclei. ${\ displaystyle {\ vec {I}}}$

Since the atomic nuclei (also the lightest, the proton ) are always composed particles, nuclear spin is not a spin in the narrower sense. ${\ displaystyle {\ vec {s}}}$

Nuclear spin is mainly used for chemical analyzes ( nuclear magnetic resonance spectroscopy ) and for medical examinations ( magnetic resonance tomography ), both due to its magnetic properties.

The term nuclear spin often only refers to its quantum number , which can assume the following values: ${\ displaystyle I}$

${\ displaystyle I = 0, \, {\ tfrac {1} {2}}, \, 1, \, {\ tfrac {3} {2}}, \, \ ldots}$

As a physical angular momentum, it then has the size

${\ displaystyle | {\ vec {I}} | = \ hbar {\ sqrt {I (I + 1)}}}$

with Planck's reduced quantum of action . ${\ displaystyle \ hbar}$

## occurrence

The nuclear spin is the sum:

${\ displaystyle {\ vec {I}} = \ sum _ {i = 1} ^ {A} ({\ vec {s}} _ {i} + {\ vec {\ ell}} _ {i})}$
• the spins (quantum number always )${\ displaystyle {\ vec {s}} _ {i}}$${\ displaystyle s = 1/2}$
• the orbital angular momentum (quantum numbers are always whole numbers )${\ displaystyle {\ vec {\ ell}} _ {i}}$${\ displaystyle \ ell}$

of the core building blocks ( : total mass number of protons and neutrons ). ${\ displaystyle A}$ ${\ displaystyle A}$

At the above For summation, the quantum mechanical rules of adding angular momentum are to be applied.

There are two options, depending on the number of core components: ${\ displaystyle A}$

Mass number ${\ displaystyle A = NZ + PZ}$ Nuclear type ( neutron number / proton number ) Nuclear spin quantum number ${\ displaystyle I}$ example annotation
odd ug or
gu
half-integer ${\ displaystyle I (_ {1} ^ {1} \ mathrm {H}) = 1/2}$
${\ displaystyle I (_ {\; 83} ^ {209} \ mathrm {Bi}) = 9/2}$
${\ displaystyle _ {1} ^ {1} \ mathrm {H}}$ is the proton
straight gg integer , in the basic state
${\ displaystyle = 0}$
${\ displaystyle I (_ {\; 6} ^ {12} \ mathrm {C}) = 0}$ In terms of energy, it is most favorable when neutrons and protons align with one another to form pairs with antiparallel angular momenta (see also Bethe-Weizsäcker formula (pairing ratio) );
Nuclei with zero nuclear spin also have no magnetic moment .
uu integer, in the basic state
${\ displaystyle \ geq 0}$
${\ displaystyle I (_ {\; \, 81} ^ {206} \ mathrm {Tl}) {\ mathord {=}} 0}$
${\ displaystyle I (_ {\; 7} ^ {14} \ mathrm {N}) = 1}$
...
${\ displaystyle I (_ {41} ^ {90} \ mathrm {Nb}) {\ mathord {=}} 8}$
Neither protons nor neutrons can completely combine to form pairs, which is why many uu nuclei also have a nuclear spin in their ground state . ${\ displaystyle I {\ mathord {>}} 0}$

Only four light UU nuclides are stable, namely , , with each and with , possibly even the very rare metastable isotope in the first excited state with . The remaining uu nuclides are metastable or unstable. There is a database for the spins of all known stable and metastable nuclides. ${\ displaystyle _ {1} ^ {2} \ mathrm {D}}$${\ displaystyle _ {3} ^ {6} \ mathrm {Li}}$${\ displaystyle _ {\; 7} ^ {14} \ mathrm {N}}$${\ displaystyle I {\ mathord {=}} 1}$${\ displaystyle _ {\; 5} ^ {10} \ mathrm {B}}$${\ displaystyle I {\ mathord {=}} 3}$ ${\ displaystyle _ {\; \; \; 73} ^ {180 \ mathrm {m}} \ mathrm {Ta}}$${\ displaystyle I {\ mathord {=}} 9}$

In excited energy levels the nuclear spin quantum number generally has different values ​​than in the ground state; However, it is always an integer for an even mass number and half-integer for an odd mass number.

The nucleus of the neutron has no electrical charge , but it does have a magnetic moment, and this is directed in the opposite direction to its nuclear spin. Therefore, the magnetic moment of an entire nucleus can be aligned antiparallel to the nuclear spin despite a positive electrical charge, e.g. B. the isotope of oxygen. ${\ displaystyle {} _ {\; 8} ^ {17} \ mathrm {O}}$

## Uses

Nuclear spin is used, or more precisely: the magnetic moment associated with it, in nuclear magnetic resonance . In the external magnetic field , the energy of the nucleus depends on how the nuclear spin (and the associated magnetic moment) is aligned with this field. With magnetic fields of a few Tesla, this results in a splitting of the energy level of the basic state of the nucleus in the order of magnitude of 10 −25 J , corresponding to a photon frequency of around 100  MHz (corresponds to a radio frequency in the ultra-short wave range ). Corresponding electromagnetic radiation can be absorbed by the atomic nuclei .

### Structural analysis

In the chemical structure analysis by nuclear magnetic resonance spectroscopy (engl. Nuclear magnetic resonance , NMR), the effects are observed, the surrounding electrons and adjacent atoms have to the nuclear spin. For example, electrons in the vicinity generate an additional magnetic field, which accordingly strengthens or weakens the external field. This shifts the frequencies at which the resonance condition is met.

### medicine

Magnetic resonance tomography or nuclear spin tomography uses nuclear magnetic resonance. Magnetic resonance tomographs in medical use usually measure the distribution of hydrogen atom nuclei (protons) in the body. In contrast to X-rays , changes in the tissue can usually be made clearly visible. For three-dimensional sectional images , magnetic fields with a gradient (i.e. a continuous increase in strength) are used so that conclusions can be drawn about the spatial position from the frequency at which the resonance condition is met.

## Macroscopic effects

As angular momentum, the nuclear spin is quantified in the same unit as the angular momentum of the shell , but because of its more than 1000 times smaller magnetic moment, it has only extremely minor effects on the magnetic properties of atoms or macroscopic pieces of matter. At very low temperatures, however, the effects of the degrees of freedom (setting options) of the nuclear spins are clearly visible in individual cases : ${\ displaystyle \ hbar}$

• The specific heat of hydrogen gas (H 2 ) shows a special temperature profile at temperatures below 100 K. This can only be explained by the fact that the two nuclei (protons) of the gas molecules each have a nuclear spin 1/2, which they put parallel in 3/4 of the molecules ( orthohydrogen ), in 1/4 of the molecules antiparallel ( parahydrogen ). In both cases, the total spin of the two nuclei (and of the molecule) is an integer, but in orthohydrogen there are no rotational levels with an uneven molecular angular momentum, in parahydrogen those with an even one. These attitudes remain in the gas molecules for weeks despite the numerous collisions with one another. This discovery was the first to prove that the proton has nuclear spin 1/2.
• The Bose-Einstein condensation , which converts liquid helium into a superfluid state, only takes place with the common isotope helium-4, but not with the rare helium-3. The reason is that a helium-4 nucleus has a nuclear spin of 0 which makes the whole atom a boson , while a helium-3 nucleus has a nuclear spin of 1/2 which makes the whole atom a fermion . This has an effect on the symmetry or antisymmetry of the quantum mechanical state of the liquid helium compared to the exchange of two atoms and leads to the described difference in the macroscopic behavior of the two isotopes.

## literature

• Jörn Bleck-Neuhaus: Elementary Particles. Modern physics from the atoms to the standard model . Springer, Heidelberg 2010, ISBN 978-3-540-85299-5 , 7.1.

## Individual evidence

1. Database on BNL.gov