Longitudinal space-charge effects in a retarding field energy analyzer

Theoretical and experimental work has been carried out to study the longitudinal space-charge effects in a retarding field energy analyzer. A one-dimensional, steady state model for both a monoenergetic beam and a thermal beam has been developed for this purpose. Potential improvements of using two-dimensional and time-dependent solutions are also briefly discussed. The study shows that, if the current density inside the device is higher than a critical value, the longitudinal space-charge effect and the formation of a potential minimum similar to the virtual cathode formation in an electron gun will distort the measured energy spectrum. The measured FWHM and the rms energy spread will be affected. The measured mean energy will also be shifted toward the low-energy side. By using a two-dimensional correction, the theoretical model also qualitatively explains the appearance of a visible tail at the high-energy side of the spectrum, as observed in experiments. According to the theory, to avoid this measurement distortion due to the longitudinal space charge, care has to be taken to limit the current density inside the device.

Figure 1

Diagram of a cylindrical energy analyzer.

Figure 2

Measured energy spectrum for a 5 keV beam, with a tail at the high-energy side.

Figure 3

One-dimensional model of the energy analyzer.

Figure 4

Potential distribution with a potential minimum. Beam energy is 5500 eV. (a) Retarding voltage at 0 V. (b) Retarding voltage at $-1000\mathrm{V}$.

Figure 5

Monotonic potential distribution. Beam energy is 5500 eV and the retarding voltage is $-3500\mathrm{V}$.

Figure 6

Potential distribution with a virtual cathode. Beam energy is 5500 eV and the retarding voltage is $-5470\mathrm{V}$. (a) Full potential range. (b) Enlarged at the vicinity of the virtual cathode.

Figure 7

Response of the energy analyzer due to the space-charge force. (a) Collector signal. (b) Beam energy spectrum.

Figure 8

Potential distribution for a beam with initial energy spread.

Figure 9

Simulation of the energy analyzer performance for $\lambda =0.062$. (a) Potential distribution with retarding voltage at $-5067\mathrm{eV}$. (b) Particle density distribution with retarding voltage at $-5067\mathrm{eV}$. (c) Energy analyzer collector signal and energy spectrum. rms energy spread of 2.2 eV, FWHM of 5.1 eV.

Figure 10

Simulation of the energy analyzer performance for $\lambda =1.2$. (a) Potential distribution with retarding voltage at $-5067\mathrm{V}$. The potential minimum is $-5073\mathrm{V}$. (b) Particle density distribution with retarding voltage at $-5067\mathrm{eV}$. (c) Energy analyzer collector signal and energy spectrum. Because of the state jump, the energy spectrum is close to a delta function.

Figure 11

Experimental setup for energy analyzer testing.

Figure 12

Comparison of the experimental results with theory. (a) Experimental results of beam energy spread for two different currents inside the device. Curve I for current of 0.2 mA ($\lambda \sim 0.062$, estimated), rms energy spread of 2.2 eV, FWHM of 3.4 eV. Curve II for current of 2.6 mA ($\lambda \sim 0.8$, estimated), rms energy spread of 3.2 eV, FWHM of 1.1 eV. (b) Simulation results of beam energy spread for different normalized current densities. Curve I for $\lambda =0.062$, rms energy spread of 2.2 eV, FWHM of 5.1 eV. Curve II for $\lambda =1.2$. Ideally, curve II should be a delta function and have an energy spread of zero. (c) Simulation results with the correction of the two-dimensional effect. Curve I for $\lambda =0.062$, rms energy spread of 2.2 eV, FWHM of 5.1 eV. Curve II for $\lambda =1.2$, rms energy spread of 5.1 eV, and FWHM of 0.49 eV.