Problems

6.2 Position-insensitive detectors

Problem 6.21. (Requires material in Section 6.5.) The magnitude of the charge pulse from a proportional counter fluctuates in value from one incident x ray to another, even when the incident x rays all have the same energy, e.g., those obtained from an 55Fe (iron 55) radioactive source. (a) Consider the detection of 6.0-keV x rays in an argon-filled proportional counter. What is the standard deviation in units of keV of these fluctuations if they arise mostly from Poisson fluctuations in the number of ion pairs created by the initial photoelectron? Assume that there are no escape photons, and consider only the first generation of ion pairs, those created by the several initial photoelectrons with a combined energy of 6.0 keV. What is the fractional energy resolution, defined as the FWHM of the response curve divided by the mean energy, at this x-ray energy? (b) What are the fractional energy resolutions at energies 2 keV and 30 keV? [Ans. ~0.4 keV, ~15%; ~25%, ~7%]

6.3 Position-sensitive detectors

Problem 6.31. Demonstrate that there is a maximum of the electric potential V that could trap a free electron within the n-type region of a CCD. Let the n-type region, of thickness d, contain only fixed positive charges with a charge density +20p0. The depleted portion of the p-type region has thickness 5d and fixed charge density —p0. The undepleted portion with its piled up free holes results in a zero electric field at x = 0 (see sketch).

depleted

Neglect the effect of any residual free charges in the depleted regions or of the electrodes or insulator. Proceed by constructing a Gaussian pillbox with its base at x = 0 (see sketch). From Gauss's law, find the electric field Ex(x) in both depleted regions and, from the results, find the potential V(x) in both regions. Set undepleted

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