## Spin

The spin of a particle is a fundamental particle property that can be treated mathematically like an angular momentum.
Different particles can be assigned different spin values s. For example, electrons, protons, and neutrons (all s=½) are called spin-½ particles, whereas a deuteron, a particle composed of proton and neutron, (s=1) we speak of a spin-1 particle. In nuclei, instead of s, one often speaks of the nuclear spin I.
Each spin-afflicted particle may occupy different energy levels associated with the orientation of the spin with respect to a preferred direction sz, most often the direction of the external magnetic field. In general, a particle can take 2s+1 orientations and thus occupy the same number of energy levels.

For example, this results in two possible orientations or energy levels for a proton, as illustrated in the graph above. As you can see, the distance between the energy levels increases with increasing magnetic field strength, which is still important for the later analysis of the polarization.

## Polarization

In nuclear physics, the term polarization of a particle ensemble means the over-occupation of one state with respect to another, or in other words: more particles are at one energy level than the other.
Mathematically, this relationship can be expressed as follows:

$P_z := \frac{ \langle s_z \rangle }{s} = \begin{cases} \frac{ N_{\frac{1}{2}} - N_{-\frac{1}{2}} }{ N_{\frac{1}{2}} + N_{-\frac{1}{2}} }\phantom{\frac{ N_{1} - N_{-1} }{ N_{1} + N_{0} + N_{-1} }} \text{for } s = \frac{1}{2}\\ \frac{ N_{1} - N_{-1} }{ N_{1} + N_{0} + N_{-1} }\phantom{\frac{ N_{\frac{1}{2}} - N_{-\frac{1}{2}} }{ N_{\frac{1}{2}} + N_{-\frac{1}{2}} }} \text{for } s = 1 \end{cases}$

where N indicates the respective number of particles at the different energy levels. The example below shows two examples.

There is always a certain over-occupancy of one state for the benefit of another. Due to the different energy levels and the endeavor of each system to minimize its energy lower energy states are preferably occupied. Therefore, for a given temperature T and magnetic field B, an equilibrium state is established after a while. In the so-called thermal equilibrium, often abbreviated to TE, the polarization can be calculated by

$P \overset{TE}{=} \tanh \left( \frac{ -g \mu_N s B }{ kT } \right)$

The graph shows the polarization at low temperatures for different particles. As one can see, the polarization increases to higher magnetic field, as well as to lower temperatures.

The method of "dynamic nucleon polarization", abbreviated DNP, allows eventually far more polarization to be achieved. For this purpose, practically the high polarization of the electrons is transferred to the nuclei.

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