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I. H. McLaren coauthored with W. C. Wiley the 1955 Bendix study of time-of-flight mass spectrometry with improved mass resolving power. A notable part of their analysis was time-lag focusing: waiting between ion formation and application of an accelerating pulse lets initial velocities become position differences that can compensate some arrival-time spread.^[ref]

A first-order calculation exposes the mechanism. Let $T(v,s)$ be arrival time after immediate extraction for initial axial velocity $v$ and position $s$. During a field-free delay $\tau$, an ion moves to $s+v\tau$. Apart from the common delay added to all times,The partial derivatives are evaluated at $v=0$ and the reference position. First-order sensitivity to $v$ vanishes ifA positive delay requires opposite signs of these sensitivities. For example, $T_v=2\times 10^{-9}{\mathrm{s}}^2{\mathrm{m}}^{-1}$ and $T_s=-10^{-3}\mathrm{s}{\mathrm{m}}^{-1}$ give $\tau =2\mu \mathrm{s}$. These values illustrate the cancellation, rather than reproducing an instrument setting from the paper.

This also reveals the tradeoff: the cancellation needs a nonzero position sensitivity, whereas ideal first-order space focusing sets $T_s=0$. The paper's delayed-extraction condition and measured spectra therefore address competing sources of broadening, not a universal elimination of every initial-condition effect.^[ref]

A mass spectrometer is an instrument that produces or receives ions, separates them according to mass-to-charge ratio, and records their signals. Its major functions are supplied by an ion source, a mass analyzer and a mass spectrometer detector. The result is a mass spectrum; interpreting a peak also requires knowing which ionic chemical species produced it.

The 1955 instrument described by W. C. Wiley and I. H. McLaren provides a concrete apparatus example. An ion source and accelerating fields launch ions into an evacuated drift region toward a collector. Arrival times separate mass-to-charge ratios; the paper analyses how initial positions and velocities broaden the recorded peaks, and uses a two-field ion-gun geometry to improve focusing.^[ref] This is one analyzer design, not a definition requiring every mass spectrometer to measure flight time.

For an ideal time-of-flight mass spectrometer, neglect initial speed and acceleration duration. Across an electric potential difference $U$, an ion of positive electric charge $q$ gains $qU=m{v}^2/2$. A field-free drift of length $L$ then givesIf one peak arrives at $40\mu \mathrm{s}$ and another at $44\mu \mathrm{s}$, identical acceleration and drift conditions imply a mass-to-charge ratio ratio $(44/40{)}^2=1.21$. It is a mass ratio only if the charge states are also equal. Peak position, width and intensity answer different questions: ionic identity, resolving ability and response-weighted abundance.

By W. C. Wiley and I. H. McLaren. Review of Scientific Instruments 26(12), 1150–1157 (December 1955). DOI: doi.org/10.1063/1.1715212

Original paper: masspec.scripps.edu/learn/ms/pdf/1955_WileyWC.pdf

Page 1150: ion source, evacuated drift tube and collector; later sections analyze initial-position/velocity broadening and two-field ion-gun focusing, then compare the analysis with recorded spectra.

Time-of-flight mass spectrometry separates ions by measuring how long they take to reach a detector after a timed acceleration pulse. The simplest instrument has an ion source, an accelerating electric field, and a drift tube kept under vacuum. For ions of equal positive electric charge accelerated through the same electric potential difference, smaller mass gives greater speed and earlier arrival.^[ref]

Let $U$ be the accelerating electric potential difference, $q$ the positive electric charge, $m$ the mass, and $L$ the field-free drift length. Neglecting initial speed and the time spent accelerating, conservation of energy givesFor an ideal $1\mathrm{m}$ drift at $1000\mathrm{V}$, a singly charged ion of mass $100\mathrm{u}$ arrives after approximately $22.8\mu \mathrm{s}$; $400\mathrm{u}$ takes $45.5\mu \mathrm{s}$. Here $\mathrm{u}$ is the unified atomic mass unit. Quadrupling mass doubles the ideal flight time. A doubly charged $200\mathrm{u}$ ion has the same mass-to-charge ratio as the singly charged $100\mathrm{u}$ ion and therefore the same ideal arrival time.

Real peaks broaden because ions begin at different positions and with different kinetic energies. In 1955, W. C. Wiley and I. H. McLaren analysed these effects and demonstrated an ion gun with two accelerating regions that reduced timing differences caused by initial position. Comparing calculated focusing conditions with measured spectra turned a timing idea into a practical improvement in mass spectrometry.^[ref]

W. C. Wiley worked on time-of-flight mass spectrometry at Bendix Aviation Corporation Research Laboratories. With I. H. McLaren, Wiley published a 1955 instrument study addressing the arrival-time spread of ions with the same mass-to-charge ratio. Their apparatus combined an ion-producing source, accelerating electric fields, an evacuated drift region and a collector; its analysis separated broadening from initial position and initial motion.^[ref]

The basic timing principle can be derived without the focusing details. An initially stationary ion of positive electric charge $q$ accelerated through an electric potential difference $V$ acquires kinetic energy $qV=m{v}^2/2$. Over a field-free drift length $L$,For a fixed charge state and instrument setting, differentiating $m\propto t^2$ gives the approximate mass resolving powerFor illustration, a $20\mu \mathrm{s}$ arrival time with a $20\mathrm{ns}$ timing width corresponds to $R\simeq 500$, if that width is the relevant small peak-width measure.

The focusing problem addressed by Wiley and I. H. McLaren is therefore substantive: a spread of initial conditions can broaden a peak even when the detector timing is precise. Their work on space focusing and time-lag focusing changes the ion trajectories to reduce that broadening, rather than merely measuring the same broad peak with a faster clock.