Polarization of an electron beam

The Stokes parameters for the polarization of an electron beam can be represented in a Cartesian basis which also provides a convenient pictorial view for the polarization state of an electron beam. Since the polarization of an ensemble of electrons requires the determination of spin projections along preselected directions, the classical vector model of a precessing spin will first be discussed. Here the spin is represented by a vector s of length 3/2 (in atomic units) which processes around a preselected direction, yielding as expectation values the projections (in atomic units, see Fig. 9.1) 

This means, the spin projection along the preselected 

If the individual electrons of a beam have different spin directions which remain constant in a statistical average, one can describe the polarization of this beam by the result of three successive measurements of the relative excess Pt of spin projections into three preselected Cartesian direction i = x, y, and z. This is demonstrated for an electron beam of six electrons with precessing spins as indicated in Fig. 9.2. The intensities Nt for electrons with (+) spin projections along the selected coordinate axes then follow from individual measurements for each of the six precessing spins  

The result reflects what one would have expected naively from looking at Fig. 9.2, in particular from Fig. 9.2(b) the spins in the +y- and — y-directions compensate, hence, there is no net effect for Py similarly, one spin along + z compensates for one spin along —z. As a result one has effectively one spin component along +z, out of six possibilities, i.e., Pz = +1/6 and one component along + x out of six possibilities, i.e., Px= +1/6. The beam is partially polarized. 

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has 

This means, the spin projection along the preselected axis (the z-axis) can be 

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Starting from the vector model of a precessing spin, the polarization properties of an electron beam with electrons of spin (l/2)h are explained and then formulated... 

Figure 9.2 Illustration of an electron beam of six electrons with different spin polarization components. It is assumed that the electrons travel along the z-axis and the spin vectors shown are constant in a statistical average, i.e., they are repeated when the beam passes by. (a) Vector model of precessing spins (b) short notation showing only the corresponding...

Recently, an interesting correlation between the laser pulse polarization and the ellipticity of the electron beam profile has been observed [71]. However, no major influence of laser polarization on the efficiency of the electron acceleration processes has been observed so far, nor this influence has been predicted by theory and simulations, differently from the proton acceleration. For proton acceleration, a great improvement on bunch charge and quality are expected by using circularly polarized laser pulses focused on thin foils at ultra-high intensities [72-74]. 

With these properties of the electron s spin, the polarization vector P for an electron beam can be defined taking into account the contributions from the individual electrons j (for simplicity the subscript j is not always shown in the... 

Another factor that influences strongly the evolution of polarization reversal in fe bulk crystals is the limited minimization of the depolarization field by means of screening charges. In the case of the afm tip of R 50 nm the screening charge is limited both by the size of the tip and its non-ohmic contact with the sample surface. Such an effect is especially pronounced in the case of indirect electron beam exposure method described in this paper. The strong limitation of the screening phenomenon provides a necessary condition for the observed effect of fdb [13-15],... 

Second, the velocity distribution function (VDF) of hot electrons was directly measured to clarify the energy deposition process using X-ray line polarization spectroscopy. When the plasma has electromagnetic field anisotropy, polarized X-rays corresponding to the magnetic quantum number are emitted. In the case of polarization spectroscopy in an electron beam ion trap (EBIT) [21], the polarization degree P is generally defined by... 

When TTF is introduced into the PSt structure and irradiated with an electron beam in the presence of CBr, some TTF units in the PSt side chain are easily converted to the radical cations via the electron transfer to CBr. The large difference in polarity between neutral species and cation radicals results in a large difference in solubility with little consequent swelling on development. 0.2 pm resolution patterns were delineated in this resist with about 10 pC/cm. ... 

The total cross section cre of an electron for x-ray scattering, given in Equation (1.38), was calculated for an unpolarized incident x-ray. Calculate the total scattering cross section for a linearly polarized incident beam. 

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