In general, the emission pattern of photoelectrons is not isotropic in space, but produces a characteristic angular distribution. In the case of gas-phase (single) photoionisation using linearly polarised light, this differential cross section is expressed in terms of an asymmetry or b parameter:
 

where A is proportional to the photoionisation cross section for a particular ionic state, E is the photon energy, P is the degree of linear polarisation, and q is the angle between the polarisation axis and the direction of the ejected electron. The variation of b with photon energy depends on the interference of the partial waves which contribute to the final channel and is therefore a sensitive probe of the photoionisation dynamics. The energy variation in  is generally gradual and b lies within the range 2 to -1, the limits corresponding to cos²q and sin²q distributions respectively. However, at certain photon energies, photoionisation can also occur indirectly via intermediate neutral states. It is well known that the interference between the two routes to ionisation can give rise to dramatic changes in both the partial cross section and the angular distribution of the photoelectrons in the vicinity of 'resonance' states - see figure 1. Therefore measurements of the differential cross sections for all possible decay channels are needed to provide a comprehensive picture of resonance photoionisation process.
 

Figure 1 Partial cross sections and b parameter measurements of the krypton 42P1/2 and 42P3/2 states at 14.000 and 14.665eV, respectively, from Flemming et al, Phys Rev A 44 (1991) 1733. The pronounced destructive interference between the direct and resonance excitation between 24.8-25.0eV is most evident in both channels, and the corresponding rapid fluctuations in the b parameter are also apparent. Six resonance states (see upper figure) have been identified in this photon energy region.

Traditionally, a photoelectron spectrometer is used to measure the count rates at q = 0^oand 90^o; their ratio is used to derive b(E). However, one fixed analyser can be used which disperses the electrons in energy, yet preserves the angle of emission. Two such systems are cylindrical mirror and toroidal analysers whose axial symmetry allows a 360^o field of view of the interaction region. This property of CMA’s and toroidal analysers is well known and enables one to measure energy-resolved  angular distributions using position-sensitive detectors. Recently, we have demonstrated how b parameters can be determined accurately using a toroidal spectrometer with a position-sensitive detectors in the angle-dispersive plane. The multi-angle detection facility of the system greatly enhances the detection efficiency, allowing the option of higher photon energy resolution or polarisation than ordinarily possible. Furthermore, the data collection procedures generally have fixed accumulation times for each energy and angle, resulting in large errors at small count rates. Such regions of low cross section are often the most interesting in terms of rapid variations in b(E) (e.g. ‘window’ resonances in rare gases - see Figure 1) so the reduction in statistical accuracy is a significant disadvantage. In our technique, a fixed number of counts - covering a wide range (~140^o) of emission angles - were accumulated at each photon energy. (A resistive anode encoder lends itself to this data accumulation method as it counts events one-at-a-time across its whole active area.) The accuracy of the measurement method is therefore independent of count rate and variations in the photon flux. However, the accumulation time and photon flux are also recorded at each photon energy so that the count rate (cross section) can be recovered later if required. Thus this method is primarily one for measuring angular distributions, whereas conventional approaches generally determine count rates at each angle.
 

In this case the detector efficiency (as a function of angle) was calibrated using known b in flat part of spectrum (eg. 25.25eV), and then applied at all the other photon energies. Then the b parameter variation was determined by fitting the equation to each recorded image using fixed polarisation (P = 0.9) - a typical image and its fit is shown (right) and the final spectra are shown and discussed in Figure 2.   Full details can be found in:        Wightman et al (1998) J Elec Spec 95 203.

Figure 2. Upper figures show the contour plots of the measured angular distributions for the and 4²P_3/2 (left) and 4²P_1/2 (right) states of krypton in the photon energy region covering the lowest 'window' resonances. They have the same intensity scale as a fixed number of points per energy were recorded, regardless of changes in the cross sections - as discussed in the text. The b parameter spectra derived from the surface plots are also shown and can be compared to those in figure 1. Although the overall agreement is excellent, the slight difference in the vertical scale is probably due to the choice/accuracy of polarisation value used in both data sets. The near constancy of the statistical error bars as a function of b is due to the new data collection procedure. In particular, this method is more reliable over conventional methods for distinguishing small variations in b upon a large b background.