[Paper Review] On Schottky Noise and Shot Noise

Xiangcheng Chen|arXiv (Cornell University)|May 29, 2018

Quantum-Dot Cellular Automata3 citations

TL;DR

This paper argues that Schottky noise and shot noise, while both arising from the discrete nature of charge carriers, are fundamentally distinct in their spectral characteristics: shot noise exhibits a flat, white power spectral density, whereas Schottky noise in circular accelerators appears as discrete harmonic spikes at multiples of the revolution frequency. The key contribution is the formal distinction between the two, challenging the widespread synonymity in accelerator physics literature.

ABSTRACT

Schottky noise is a common term widely acknowledged by the community of accelerators, especially of circular machines. It is referred to the incoherent signal arising from the circulating charged particles in the accelerator. The noise is named after W. Schottky, who first discovered a then new type of noises in direct electric current and called it shot noise himself. Because of this, the Schottky noise is considered as an alias for the shot noise in accelerator literatures. But is it really so? In this essay I compare the two noises side-by-side by formulas and argue that even though they arise from the same discrete nature of charge carriers, the noise spectral patterns are inherently distinct, whereby the synonymity of these two terms is questionable.

Motivation & Objective

• To clarify the conceptual and mathematical distinction between Schottky noise and shot noise, which are often treated as synonymous in accelerator physics.
• To address the misconception that Schottky noise is merely a form of shot noise, despite their shared origin in discrete charge carriers.
• To demonstrate through spectral analysis that the two noises exhibit fundamentally different power spectral density patterns.
• To argue that the term 'Schottky noise' should be preserved as a distinct concept due to its unique diagnostic utility in beam instrumentation.
• To provide a rigorous derivation of the power spectral density for both noise types using statistical and Fourier analysis.

Proposed method

• Modeling shot noise as a stochastic stream of point charges with random arrival times, using Dirac delta functions to represent individual charge pulses.
• Applying the Wiener-Khinchin theorem to compute the power spectral density (PSD) of the beam current, linking autocorrelation to Fourier transform.
• Deriving the autocorrelation function for Schottky noise in a coasting beam, accounting for periodic repetition of each particle's transit at the detector.
• Calculating the PSD of Schottky noise as a sum of Dirac delta functions at harmonics of the revolution frequency $ f_r $, with amplitude $ 2q\bar{I}f_r $ per harmonic.
• Using ergodicity and ensemble averaging to simplify time-averaged correlations, particularly for uniformly distributed initial phases.
• Comparing the resulting spectral patterns: flat white noise for shot noise vs. discrete harmonic spikes for Schottky noise.

Experimental results

Research questions

• RQ1 Is Schottky noise truly equivalent to shot noise, given their shared origin in discrete charge carriers?
• RQ2 How do the power spectral densities of shot noise and Schottky noise differ in frequency domain characteristics?
• RQ3 Why does Schottky noise exhibit a harmonic spike structure while shot noise is flat and white?
• RQ4 What physical and mathematical conditions lead to the transformation of white noise into discrete spectral lines in circular accelerators?
• RQ5 To what extent does the spectral distinction justify retaining 'Schottky noise' as a distinct term in accelerator physics?

Key findings

• The power spectral density of shot noise is flat and white, given by $ S_o(f) = 2q\bar{I} $, independent of frequency.
• The power spectral density of Schottky noise consists of discrete Dirac delta functions at integer multiples of the revolution frequency $ f_r $, with $ S_c(f) = 2q\bar{I}f_r \sum_{k=1}^{\infty} \delta(f - kf_r) $.
• The total power in any frequency band of width $ f_r $ is identical for both noises, equaling $ 2q\bar{I}f_r $, indicating conservation of total noise power.
• The harmonic structure of Schottky noise arises from the periodic repetition of particle transit, which folds the continuous spectrum into discrete lines.
• When momentum spread is introduced, the spectral lines broaden into bands, with height decreasing inversely with harmonic number to preserve total power.
• Despite shared origin in charge quantization, the spectral patterns are inherently different, undermining the synonymity of 'Schottky noise' and 'shot noise'.