Sherman function

Correction to Coulomb potential due to spin-orbit coupling. The Coulomb potential, result of the interaction of the electron with the charged nucleus is shown in green. The new potential for spin up (blue) and spin down (red) electrons are shown after spin-orbit correctios have been taken into account

When an electron beam is polarized, an unbalance between spin-up, ${\displaystyle n_{up}}$, and spin-down electrons, ${\displaystyle n_{down}}$, exists. The unbalance can be evaluated through the polarization ${\displaystyle P}$ [4] defined as

${\displaystyle P={\frac {n_{up}-n_{down}}{n_{up}+n_{down}}}}$.

It is known that, when an electron collides against a nucleus, the scattering event is governed by Coulomb interaction. This is the leading term in the Hamiltonian, but a correction due to spin-orbit coupling can be taken into account and the effect on the Hamiltonian can be evaluated with the perturbation theory. Spin orbit interaction can be evaluated, in the rest reference frame of the electron, as the result of the interaction of the spin magnetic moment of the electron

${\displaystyle {\boldsymbol {\mu }}_{S}=-g_{\text{s}}\mu _{\text{B}}{\frac {\mathbf {S} }{\hbar }},}$

with the magnetic field that the electron sees, due to its orbital motion around the nucleus, whose expression in the non-relativistic limit is:

${\displaystyle \mathbf {B} ={\frac {1}{m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} .}$

In these expressions ${\displaystyle \mathbf {S} }$ is the spin angular-momentum, ${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton, ${\displaystyle g_{\text{s}}}$ is the g-factor, ${\displaystyle \hbar }$ is the reduced Planck constant, ${\displaystyle m_{\text{e}}}$ is the electron mass, ${\displaystyle e}$ is the elementary charge, ${\displaystyle c}$ is the speed of light, ${\displaystyle U=eV}$ is the potential energy of the electron and ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$ is the angular momentum.

Due to spin orbit coupling, a new term will appear in the Hamiltonian, whose expression is[5]

${\displaystyle V_{\text{SO}}={\boldsymbol {\mu _{s}}}\cdot \mathbf {B} }$.

Due to this effect, electrons will be scattered with different probabilities at different angles. Since the spin-orbit coupling is enhanced when the involved nuclei possess a high atomic number Z, the target is usually made of heavy metals, such as mercury,[1] gold[6] and thorium.[7]

If we place two detectors at the same angle from the target, one on the right and one on the left, they will generally measure a different number of electrons ${\displaystyle n_{R}}$ and ${\displaystyle n_{L}}$. Consequently it is possible to define the asymmetry ${\displaystyle A}$, as[2]

${\displaystyle A={\frac {n_{R}-n_{L}}{n_{R}+n_{L}}}}$.

Sherman function as a function of the angle for mercury (Z = 80) with β = 0.2. Typically detectors are placed in a position where the effect is maximized, 120° for gold and mercury[1]

The Sherman function ${\displaystyle S(\theta )}$ is a measure of the probability of a spin-up electron to be scattered, at a specific angle ${\displaystyle \theta }$, to the right or to the left of the target, due to spin-orbit coupling.[8][9] It can assume values ranging from -1 (spin-up electron is scattered with 100% probability to the left of the target) to +1 (spin-up electron is scattered with 100% probability to the right of the target). The value of the Sherman function depends on the energy of the incoming electron, evaluated via the parameter ${\displaystyle \beta ={\frac {v}{c}}}$.[1] When ${\displaystyle S(\theta )=0}$, spin-up electrons will be scattered with the same probability to the right and to the left of the target.[1]

Then it is possible to write

${\displaystyle n_{R}={\frac {n_{up}[1+S(\theta )]+n_{down}[1-S(\theta )]}{2}}}$

${\displaystyle n_{L}={\frac {n_{up}[1-S(\theta )]+n_{down}[1+S(\theta )]}{2}}.}$

Plugging these formulas inside the definition of asymmetry, it is possible to obtain a simple expression for the evaluation of the asymmetry at a specific angle ${\displaystyle \theta }$,[10] i.e.:

${\displaystyle A=PS(\theta )}$.

Theoretical calculations are available for different atomic targets[1][11] and for a specific target, as a function of the angle.[8]

In the panel it is shown an example of the working principle of a Mott detector, supposing a value for ${\displaystyle S(\theta )=0.5}$. If an electron beam with a 3:1 ratio of spin-up over spin-down electrons collide with the target, it will be splitted with a ratio 5:3, according to previous equation, with an asymmetry of 25%.