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A non‐Markovian model of avalanche gain statistics for a solid‐state photomultiplier
1.M. D. Petroff, M. G. Stapelbroek, and W. A. Kleinhans, U.S. Patent Number 4, 586, 068; filed Oct. 7, 1983; granted Apr. 29, 1986.
2.M. D. Petroff, M. G. Stapelbroek, and W. A. Kleinhans, Appl. Phys. Lett. 51, 406 (1987).
3.B. L. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, New York, 1984).
4.G. E. Stillman and C. M. Wolfe, in Semiconductors and Semimetals, edited by R. K. Willardson and A. C. Beer, Vol. 12 (Academic, New York, 1977), p. 291.
5.S. Karlin, A First Course in Stochastic Processes (Academic, New York, 1968), Chap. 11; N. van Kampen, Stochastic Processes in Physics and Chemistry (North‐Holland, New York, 1981), p. 72.
6.W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd ed. (Wiley, New York, 1968), p. 450; E. C. Zaehmanoglou and D. W. Thoe, Introduction to Partial Differential Equations (Wilkins and Wilkins, Baltimore, MD, 1976), p. 92.
7.C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983);
7.L. Reggiani, in Hot‐Electron Transport in Semiconductors, edited by L. Reggiani (Springer, New York, 1985), p. 7.
8.B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982), Chap. 4.
9.The integration was performed by the extended trapezoidal rule in conjunction with a cubic spline to interpolate between the data points, cf. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge, Univ., New York, 1986), Chaps. 3 and 4.
10.The iterative integration over the high‐field region was performed by the extended trapezoidal rule employing 2048 points. Accuracy was monitored partly by noticing that the solution obtained did not differ significantly from the solution obtained with 1024 points, cf. D. Kershaw, in Numerical Solution of Integral Equations, edited by L. M. Delves and J. Walsh (Clarendon, Oxford, 1974), p. 140.
11.For example, another approach employs the maximum entropy method. cf.
11.J. M. Einbu, IEEE Trans. Inf. Theory IT‐23, 772 (1977).
12.H. Cramer, Mathematical Methods of Statistics (Princeton, New Jersey, 1946), p. 227.
13.No oscillations (as a function of position r) were found in the moments calculated from Eq. (7). However, other non‐Markovian models for α (not necessarily for the SSPM) may display transient oscillations in their moments, cf. R. A. LaViolette, J. Chem. Phys. (in press).
14.M. D. Petroff (private communication).
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