General Description

The cross-section of the Møller scattering $\vec{e^-} + \vec{e^-} \rightarrow e^- + e^-$ depends on the beam and target polarizations ${\cal P}_{beam}$ and ${\cal P}_{target}$ as:

$$\sigma \propto ( 1 + \sum_{i=X,Y,Z} (A_{ii}\cdot{}{\cal P}_{targ~i}\cdot{}{\cal P}_{beam~i})) ,$$ (B1)

where $i = X,Y,Z$ defines the projections of the polarizations. The analyzing power $A$ depends on the scattering angle in the CM frame $\theta{}_{CM}$. Assuming that the beam direction is along the Z-axis and that the scattering happens in the ZX plane:

$$A_{ZZ} = -\frac{\sin^2\theta{}_{CM}\cdot(7+\cos^2\theta{}... ...heta{}_{CM}} {(3 + \cos^2\theta{}_{CM})^2}, A_{YY} = -A_{XX}$$ (B2)

The analyzing power does not depend on the beam energy. At $\theta{}_{CM}=90^o$ the analyzing power has its maximum $A_{ZZ~max}=7/9$. A transverse polarization also leads to an asymmetry, though the analyzing power is lower: $A_{XX~max}=A_{ZZ}/7$. The main purpose of the polarimeter is to measure the longitudinal component of the beam polarization.

The Møller polarimeter of Hall A detects pairs of scattered electrons in a range of $75^o < \theta{}_{CM} < 105^o$. The average analyzing power is about $<A_{ZZ}>=0.76$.

The target consists of a thin magnetically saturated ferromagnetic foil. In such a material about 2 electrons per atom can be polarized. An average electron polarization of about 8% can be obtained. In Hall A Møller polarimeter the foil is magnetized along its plane and can be tilted at angles $20-160^o$ to the beam. The effective target polarization is ${\cal P}_{target}={\cal P}_{foil}\cdot{}\cos\theta{}_{target}$.

The secondary electron pairs pass through a magnetic spectrometer which selects particles in a certain kinematic region. Two electrons are detected with a two-arm detector and the coincidence counting rate of the two arms is measured.

The beam longitudinal polarization is measured as:

$${\cal P}_{beam~Z} = \frac{N_{+}-N_{-}}{N_{+}+N_{-}}\cdot{} ... ...}{{\cal P}_{foil}\cdot{}\cos\theta{}_{target}\cdot{}<A_{ZZ}>},$$ (B3)

where $N_{+}$ and $N_{-}$ are the measured counting rates with two opposite mutual orientation of the beam and target polarizations, while $<A_{ZZ}>$ is obtained using Monte-Carlo calculation of the Møller spectrometer acceptance, ${\cal P}_{foil}$ is derived from special magnetization measurements of the foil samples and $\theta{}_{target}$ is measured using a scale, engraved on the target holder and seen with a TV camera, and also using the counting rates measured at different target angles.

The target is rotated in the horizontal plane. The beam polarization may have a horizontal transverse component, which would interact with the horizontal transverse component of the target polarization. The way to cancel the influence of the transverse component is to take an average of the asymmetries measured at 2 complimentary target angles, say 25 and $155^o$.