Hemispherical electron energy analyzer

Hemispherical electron energy analyzer

A hemispherical electron energy analyzer or hemispherical deflection analyzer is a type of detector generally used for applications where high resolution of electron energy is needed, for example different varieties of electron spectroscopy such as X-ray photoelectron spectroscopy and Auger electron spectroscopy. They can also be used in imaging applications such as photoemission electron microscopy and low-energy electron microscopy.

Function

An ideal hemispherical analyzer consists of two concentric hemispherical electrodes (inner and outer hemispheres) held at proper voltages. It is possible to demonstrate that in such a system, (i) the electrons are linearly dispersed along the direction connecting the entrance and the exit slit, depending on their kinetic energy, while (ii) electrons with the same energy are first-order focused.

When two potentials, $V_{1}$ and $V_{2}$, are applied to the inner and outer hemispheres, respectively, the electric potential and field in the region between the two electrodes can be calculated by solving the Laplace equation:

$V(r)=-\left[{\frac {(V_{2}-V_{1})}{(R_{2}-R_{1})}}\right]\cdot {\frac {(R_{1}R_{2})}{r}}+const.$

$|E(r)|=-\left[{\frac {(V_{2}-V_{1})}{(R_{2}-R_{1})}}\right]\cdot {\frac {(R_{1}R_{2})}{r^{2}}}$

where $R_{1}$ and $R_{2}$ are the radii of the two hemispheres. In order for the electrons with kinetic energy $E_{0}$ to follow a circular trajectory of radius $R_{0}={\frac {(R1+R2)}{2}}$, the force exerted by the electric field ($F_{E}=-e|E(r)|$) acts as the centripetal force ($F_{C}$) along the whole semicircular path. After some algebra, the following expression can be derived for the potential:

$V(r)=\left({\frac {V_{0}R_{0}}{r}}\right)+const.$,

where $V_{0}={\frac {E_{0}}{e}}$ is the energy of the electrons expressed in eV. From this equation, we can calculate the potential difference between the two hemispheres, which is given by:

$V_{2}-V_{1}=V_{0}R_{0}\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right)$.

The latter equation can be used to determine the potentials to be applied to the hemispheres in order to select electrons with energy $E_{0}=|e|V_{0}$, the so-called pass energy.

In fact, only the electrons with energy $E_{0}$ impinging normal to the entrance slit of the analyzer describe a trajectory of radius $R_{0}=(R_{1}+R_{2})/2$ and reach the exit slit, where they are revealed by the detector.

The instrumental energy resolution of the device depends both on the geometrical parameters of the analyzer and on the angular divergence of the incoming photoelectrons:

$\Delta E=E_{0}\left({\frac {w}{2R_{0}}}+{\frac {\alpha ^{2}}{4}}\right)$,

where $w$ is the average width of the two slits, and $\alpha$ is the incidence angle of the incoming photoelectrons. Though the resolution improves with increasing $R_{0}$, technical problems related to the size of the analyzer put a limit on the actual value of $R_{0}$. Although a low pass energy $E_{0}$ improves the resolution, the electron transmission probability is reduced at low pass energy, and the signal-to-noise ratio deteriorates, accordingly. The electrostatic lenses in front of the analyzer have two main purposes: they collect and focus the incoming photoelectrons into the entrance slit of the analyzer, and they decelerate the electrons to the kinetic energy $E_{0}$, in order to increase the resolution.

When acquiring spectra in sweep (or scanning) mode, the voltages of the two hemispheres $V_{1}$ and $V_{2}$ - and hence the pass energy- are held fixed; at the same time, the voltage applied to the electrostatic lenses is swept in such a way that each channel counts electrons with the selected kinetic energy for the selected amount of time. In order to reduce the acquisition time per spectrum, the so-called snapshot (or fixed) mode can be used. This mode exploits the relation between the kinetic energy of a photoelectron and its position inside the detector. If the detector energy range is wide enough, and if the photoemission signal collected from all the channels is sufficiently strong, the photoemission spectrum can be obtained in one single shot from the image of the detector.

See also

• Mass spectrometry