###### Tensor Spin Observables

Since the advent of nuclear physics, all experimental results have come from only three types of measurements: decay rates, cross-sections, and spin asymmetries. Asymmetries are a ratio of cross-sections where either the beam or target (or both) have their spins aligned either with (spin-up) or against (spin-down) an external magnetic field. For example,

where σ↑(↓) are the cross-sections with spin-up (spin-down).

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Tensor polarization introduces a new type measurement called a 'tensor asymmetry', described by

where σT(u) are the cross-sections with tensor polarization (T) and without polarization (unpolarized, or u).

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Because of this unique measurement, we gain access to a variety of observables that are difficult or impossible to measure with any previous technique.

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Currently, the Long Lab is leading two experiments to measure three of these tensor observables at Jefferson Lab: the tensor asymmetry Azz, the tensor analyzing power T20, and the tensor structure function b1.

###### What Is Tensor Polarization?

To talk about tensor polarization, it's helpful to first describe vector polarization, which is often just called "polarization". Due to quantum mechanics, spin-1/2 particles such as the proton have only two possible spin states called 'spin-up' and 'spin-down'. Spin-up occurs when the direction of the spin vector aligns with an external magnetic field, and spin-down occurs when the spin vector anti-aligns with the field. Vector polarization compares the number of particles in the spin-up state to the spin-down state by

Tensor polarization does not exist for spin-1/2 particles. It requires at least spin-1 systems, one of the simplest of which is a proton and neutron bound together. We call this system the 'deuteron' (deutero- for 'two' and -on for 'thing'). Again due to quantum mechanics, spin-1 particles have a spin-up and spin-down state like spin-1/2 particles, but they also have a spin-zero state. Uniquely for the deuteron, the spin-up and spin-down states (m=±1) have a different 3-dimensional structure than the spin-down (m=0) state. The animation below shows the m=±1 distribution on the left, which looks similar to a 'peanut', and the m=0 state on the right, which looks similar to a 'doughnut'.

Roughly speaking, tensor polarization is a measure of whether you have more nuclei in the 'peanut' states or in the 'doughnut' state. It's given by the equation below, which is also diagrammed on the right:

The extra '2' comes from the fact that there are two types of 'peanut' states, but only one 'doughnut' state. For Pzz=0 to be unpolarized, which means there are an equal number of up, down, and zero states, the 2 is needed for normalization. If 0<Pzz≤1, there are more 'peanuts' than 'doughnuts'. If -2≤Pzz<0, there are more 'doughnuts' than 'peanuts'.