--- title: Mass Spectrometry module: Magnetic Field moduleNumber: 6 lessonNumber: 5 order: 605 summary: > To weigh a single atom you cannot use a scale, so you use a magnetic field instead. A charged ion of unknown mass bends in a field by an amount that depends on its momentum and charge, so if every ion enters with the same velocity, its landing position reads off its mass-to-charge ratio directly. We build the instrument in two stages: crossed electric and magnetic fields that pass only ions with $v=E/B$, and a magnetic sector that bends the survivors along $r=mv/(|q|B)$. Then we ask what blurs a spectral line and how reference ions turn a position into a calibrated mass. topics: [Magnetic Field] draft: false sources: - book: Tipler & Mosca ref: "Ch. 26 — The Magnetic Field; §26-2 Motion of a Point Charge in a Magnetic Field" --- ## Crossed fields and selector geometry A velocity selector sets a transmitted speed. An electric field and a magnetic field lie perpendicular to each other and to the nominal beam direction. Particles with unequal transverse electric and magnetic forces miss the exit aperture. A narrow speed band can therefore be passed from a source whose ions have a broad initial energy distribution. Mass separation belongs to a later dispersive stage. Take a right-handed coordinate system with the beam entering along $+x$. Let the electric field point along $-y$ and the magnetic field point along $+z$. A positive ion then has an electric force toward $-y$ and a magnetic force toward $+y$. The transverse force is $$ F_y=q(vB-E). $$ The sign reverses for a negative particle, but both electric and magnetic terms reverse together. The condition for zero transverse force therefore has the same speed for either sign of charge: $$ v_s=\frac{E}{B}. $$ Here $E$ and $B$ are magnitudes. The selected speed is independent of particle mass and charge magnitude. An ion with that speed passes straight through even when its charge state differs from another accepted ion. > **Worked example (Setting a velocity selector).** A selector uses parallel plates > $d=1.0\ \mathrm{cm}$ apart at $U=1.2\ \mathrm{kV}$, giving field > $E=U/d=1.2\times10^5\ \mathrm{V\,m^{-1}}$. What magnetic field passes ions at > $v=4.0\times10^5\ \mathrm{m\,s^{-1}}$? Balancing the transverse forces $qE=qvB$, > > $$ > B=\frac{E}{v}=\frac{1.2\times10^5}{4.0\times10^5}=0.30\ \mathrm{T}. > $$ > > Neither mass nor charge appears: every ion at $4.0\times10^5\ \mathrm{m\,s^{-1}}$ > passes regardless of species, which is why a dispersive stage must follow. $$ % caption: Crossed-field velocity selector for a positive ion. With the beam moving right and the magnetic field out of the page, the electric force points down and the magnetic force up; only the ion whose speed makes the two forces equal, $v=E/B$, travels straight through. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,fill=black!8] (-2.7,1.0) rectangle (2.7,1.28); \draw[black,fill=black!8] (-2.7,-1.28) rectangle (2.7,-1.0); \node[above] at (0,1.28) {upper plate}; \node[below] at (0,-1.28) {lower plate}; \draw[acc,->,very thick] (-2.3,0)--(2.3,0); \node[acc,below] at (2.05,-0.05) {beam}; \filldraw[draw=black,fill=black!8] (-1.4,0) circle (0.11); \draw[black,->,very thick] (-1.4,0)--(-1.4,0.72); \node[black,left] at (-1.45,0.5) {mag. force}; \draw[black,->,very thick] (-1.4,0)--(-1.4,-0.72); \node[black,left] at (-1.45,-0.5) {elec. force}; \draw[black] (1.4,0.55) circle (0.16); \filldraw[black] (1.4,0.55) circle (0.03); \node[black,right] at (1.6,0.55) {$B$ out}; \end{tikzpicture} $$ The physical plate voltage and plate spacing determine the electric field only after the central gap is known. With an approximately uniform gap $d$ and a plate potential difference $V_p$, $$ E\approx\frac{V_p}{d}, \qquad v_s\approx\frac{V_p}{dB}. $$ The approximation excludes the fringing region near the plate ends. The central-field value is often adequate for an instrument with a long plate region and a beam that remains far from the edges. A short selector or a wide beam samples nonuniform electric field, so the selected speed becomes position dependent. The selector response follows directly from the force balance. Let $v=v_s+\delta v$ while the fields remain fixed. Since $E=v_sB$, $$ F_y=qB\,\delta v. $$ A faster positive ion curves in the magnetic-force direction. A slower positive ion curves in the electric-force direction. Negative ions curve oppositely in space, yet the magnitude of their departure from the central path is governed by $|q|B|\delta v|$. The charge sign must be retained when predicting which side aperture blocks the beam. The mass dependence enters when a residual force has time to act. For selector length $\ell$, a subsequent drift distance $D$, and a paraxial trajectory with negligible change in $v_x$, the vertical coordinate at a downstream aperture is $$ y=\frac{q(E-vB)}{mv^2} \left(\frac{\ell^2}{2}+\ell D\right). $$ The formula gives a first design estimate. It assumes uniform fields, small deflection angle, and a particle entering at the central line with zero transverse velocity. Light ions are rejected more strongly than heavy ions with the same velocity offset because they acquire greater transverse acceleration. A real selector therefore has an acceptance function that depends on velocity, charge state, mass, initial angle, and aperture geometry. An exit slit of width $w$ accepts the portion of phase space satisfying $|y|=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (-3.2,0)--(3.3,0) node[right] {speed change}; \draw[->,black] (0,-0.1)--(0,2.9) node[above] {transmission}; \draw[acc,very thick] plot[domain=-2.8:2.8,samples=160] ({\x},{2.4*exp(-pow(\x/1.0,2))}); \draw[dashed,black] (-1.0,0)--(-1.0,1.45); \draw[dashed,black] (1.0,0)--(1.0,1.45); \node[black,below] at (-1.0,0) {low edge}; \node[black,below] at (1.0,0) {high edge}; \node[above right] at (0.1,2.4) {selected speed}; \end{tikzpicture} $$ Beam alignment is checked with the magnetic field disabled and then with the electric field disabled. Each single-field configuration should bend a test beam in the predicted direction. Reversing the electric plate polarity and reversing the magnetic field provide independent sign checks. A centered beam with both fields present can otherwise conceal two wrong polarity assignments that happen to compensate at one operating point. ## Beam deflection and charge-to-mass inference An electric-deflection stage measures $q/m$ through transverse acceleration and screen displacement. A collimated beam enters parallel plates of length $\ell$ with speed $v$. A uniform transverse electric field $E_y$ acts only between the plates. The horizontal speed remains approximately $v$ while the transverse acceleration is $$ a_y=\frac{qE_y}{m}. $$ The time spent between the plates is $\ell/v$. At the plate exit, the transverse velocity and displacement are $$ v_y=\frac{qE_y\ell}{mv}, \qquad y_1=\frac{qE_y\ell^2}{2mv^2}. $$ After the plates, the beam drifts a distance $D$ with that transverse velocity. The total detector displacement is $$ y=\frac{qE_y}{mv^2} \left(\frac{\ell^2}{2}+\ell D\right). $$ The sign of $y$ gives the sign of $q$ once the plate polarity and coordinate system are known. A separate velocity selector sets $v=E_s/B_s$. Combining the two measurements produces $$ \frac{q}{m} =\frac{y v^2} {E_y\left(\ell^2/2+\ell D\right)}. $$ The Thomson-style crossed-field measurement combines the two observables. The selector determines speed, and electrostatic deflection determines charge-to-mass ratio. The drift term $\ell D$ often dominates screen displacement, so the plate-to-screen distance needs the same care as the plate length. $$ % caption: Electric deflection for charge-to-mass inference. The beam gains transverse displacement between the plates of length $\ell$, then travels straight over the drift distance $D$ to the screen; a separate velocity selector fixes the speed needed to solve for $q/m$. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,fill=black!8] (-2.7,0.72) rectangle (-0.9,0.9); \draw[black,fill=black!8] (-2.7,-0.9) rectangle (-0.9,-0.72); \node[above] at (-1.8,0.9) {plates}; \draw[black,very thick] (2.5,-1.25)--(2.5,1.25); \node[black,above] at (2.5,1.25) {screen}; \draw[acc,very thick,->] (-3.3,0)--(-2.7,0); \draw[acc,very thick] plot[domain=-2.7:-0.9,samples=40] ({\x},{0.30*((\x+2.7)/1.8)^2}); \draw[acc,very thick,->] (-0.9,0.30)--(2.5,1.05); \filldraw[acc] (2.5,1.05) circle (1.8pt); \draw[dashed,black] (-0.9,0)--(2.5,0); \draw[black,<->] (-2.7,-1.12)--(-0.9,-1.12); \node[black,below] at (-1.8,-1.12) {plate length}; \draw[black,<->] (-0.9,-1.12)--(2.5,-1.12); \node[black,below] at (0.8,-1.12) {drift}; \node[acc,below] at (-3.0,-0.05) {beam}; \end{tikzpicture} $$ The paraxial model fails when the deflection angle is large. In that case the time within the plates differs from $\ell/v$ because longitudinal and transverse motion couple through the actual path. Fringe fields extend the effective plate length, and a finite beam width samples different electric fields. These effects are reduced by operating at small angle, using long uniform plates, mapping the field, and calibrating displacement with a reference beam. Detector response must also be separated from trajectory geometry. A phosphor screen, position-sensitive detector, or swept collector converts an impact coordinate into a readout coordinate. Screen distortion, pixel pitch, point-spread width, and readout nonlinearity all affect the inferred $y$. Known reference positions establish the pixel-to-distance relation, including screen distortion, pixel pitch, and readout nonlinearity. An independent magnetic-deflection run tests the same quantities by a different route. A beam with known $q/m$ must have magnetic curvature consistent with the selected velocity. A speed error, field-map error, electric-plate fringe correction, detector calibration, or unrecognized charge state can produce a disagreement. Reliability improves when each stage constrains a different combination of the same physical quantities. ## Magnetic analyzers and mass-to-charge A magnetic sector analyzer separates a beam by momentum-to-charge ratio. The required circular-motion relation is used here as a measurement equation: $$ p=|q|Br, \qquad \frac{p}{|q|}=Br. $$ The radius $r$ is the fitted trajectory radius in the analyzing field. The formula concerns the momentum component perpendicular to the field. An ion with a substantial parallel velocity component follows a helical path outside planar-sector geometry. Collimation and field alignment are therefore part of the measurement definition. A beam entering a uniform sector at known speed $v$ perpendicular to $\vec B$, $$ \frac{m}{|q|}=\frac{Br}{v}. $$ In a selector–analyzer chain, $v=E_s/B_s$ at the selected speed. Let $B_a$ be the analyzer field and $r$ the fitted sector radius. Then $$ \frac{m}{|q|} =\frac{B_aB_s r}{E_s}. $$ The stages have separate measurement roles. Crossed fields set the speed. The magnetic sector converts mass-to-charge ratio into position. With unknown charge state, the reported observable is $m/|q|$. $$ % caption: Semicircular ion path in a magnetic sector analyzer. An ion entering perpendicular to the field bends on a circle of radius $r=mv/(|q|B)$; at a common selected speed a larger mass-to-charge ratio traces a larger radius and lands farther along the detector plane. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \foreach \x in {-1.4,-0.4,0.6,1.6} \foreach \y in {0.7,1.7} { \filldraw[black] (\x,\y) circle (0.03); } \node[black] at (2.8,2.3) {$B$ out}; \draw[very thick,->] (-2.2,-0.95)--(-2.2,-0.1); \node[left] at (-2.28,-0.55) {ion in}; \draw[acc,very thick,->] (-2.2,0) arc(180:0:1.9); \draw[acc!60!black,very thick,->] (-2.2,0) arc(180:0:2.5); \draw[dashed,black] (-2.2,0)--(3.0,0); \draw[black,<->] (-0.3,0.06)--(-0.3,1.84); \node[black,right] at (-0.22,0.95) {radius r}; \filldraw[acc] (1.6,0) circle (2pt); \filldraw[acc!60!black] (2.8,0) circle (2pt); \node[black,below] at (1.6,-0.12) {small m}; \node[black,below] at (2.9,-0.12) {large m}; \node[black,above left] at (-2.2,0.05) {slit}; \end{tikzpicture} $$ Convert detector coordinate into a geometric radius. A sector with a nominal centre and entrance angle has a position mapping that can be calculated from the mechanical drawing, while mounting tolerances and field-edge geometry shift the realized mapping. Reference ions with known $m/q$ establish the actual radius-to-detector relation. A broad detector may require a polynomial calibration; a narrow photographic plate can often be treated locally as linear. Acceleration-based mass analysis establishes ion energy with a source voltage, then uses a magnetic sector. An ion accelerated from rest through a potential magnitude $V$ has $$ \frac12 mv^2=|q|V. $$ Combining this energy equation with $r=mv/(|q|B)$ yields $$ \frac{m}{|q|} =\frac{B^2r^2}{2V}. $$ The measured quantity is $m/q$. The acceleration voltage controls kinetic energy per charge. Doubly charged ions gain twice the kinetic energy of singly charged ions through the same voltage, and their curvature differs accordingly. > **Worked example (Acceleration-based analyzer radius).** A singly charged ion of > mass $20\ \mathrm{u}$ is accelerated from rest through $V=1.00\ \mathrm{kV}$ and > enters a sector field $B=0.200\ \mathrm{T}$. With > $m=20(1.66\times10^{-27})=3.32\times10^{-26}\ \mathrm{kg}$ and > $q=1.602\times10^{-19}\ \mathrm{C}$, the speed after acceleration is > > $$ > v=\sqrt{\frac{2qV}{m}} > =\sqrt{\frac{2(1.602\times10^{-19})(1000)}{3.32\times10^{-26}}} > =9.8\times10^4\ \mathrm{m\,s^{-1}}, > $$ > > and the sector radius is > > $$ > r=\frac{mv}{qB}=\frac{(3.32\times10^{-26})(9.8\times10^4)}{(1.602\times10^{-19})(0.200)} > =0.102\ \mathrm{m}. > $$ > > A 10 cm radius. Because $r=\frac{1}{B}\sqrt{2Vm/q}$, a mass-$80\ \mathrm{u}$ ion at > the same $V$ and $B$ lands at twice the radius, $\sqrt{4}$, and doubling $V$ spreads > every radius by $\sqrt2$. A velocity-selected beam has an analyzer radius that increases linearly with $m/|q|$. For an acceleration-based beam, radius is proportional to $\sqrt{m/|q|}$ at fixed $B$ and $V$. The geometry therefore sets the natural detector scale. A detector with equal spatial bins has equal $m/q$ bins only after the appropriate nonlinear conversion is applied. The trajectory depends on the bending field sampled along its path. Field variation across the mechanical sector and fringing at pole edges change the effective sector angle and focusing. High-accuracy instruments map the field or use reference masses across the detector range so that calibration captures the realized magnetic geometry. $$ % caption: Stages of a selected-beam mass spectrometer. The source and slit form the beam, crossed fields pass a single speed, the magnetic sector disperses by mass-to-charge, and the detector records position and count; each stage carries its own acceptance limit. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \foreach \i/\lab in {0/source,1/slit,2/selector,3/sector,4/detector} { \draw[black,fill=black!8] ({\i*1.9},-0.45) rectangle ({\i*1.9+1.5},0.45); \node at ({\i*1.9+0.75},0) {\lab}; } \foreach \i in {0,1,2,3} { \draw[very thick,->] ({\i*1.9+1.5},0)--({\i*1.9+1.9},0); } \draw[very thick,->] (9.1,0)--(9.6,0); \node[black,below] at (4.55,-0.55) {one speed}; \node[black,below] at (6.45,-0.55) {mass sort}; \node[black,below] at (8.35,-0.55) {count}; \end{tikzpicture} $$ Beam current is lost at every aperture. The source creates an emittance in position and angle; the selector accepts a velocity interval; entrance and exit slits accept a spatial interval; the sector accepts a momentum interval. Reducing each interval raises resolution and reduces signal. Required resolving power, detector noise floor, source intensity, and available acquisition time determine the operating point. ## Charge state, energy, and beam identity Ion charge is written $q=ze$, where $z$ is an integer charge state and $e$ is the elementary-charge magnitude. A magnetic analyzer measures $m/z$ when charge is expressed in units of $e$. A peak at a given $m/z$ can arise from a light singly charged ion, a heavier multiply charged ion, or a molecular fragment. Peak identity therefore combines mass analysis with source chemistry, isotopic patterns, charge-state knowledge, and sometimes independent energy measurements. At fixed selected velocity, a sector radius satisfies $$ r=\frac{m}{|q|}\frac{E_s}{B_sB_a}. $$ Ions with equal $m/z$ follow the same ideal trajectory even when their masses and charge states differ. A charge-state distribution can therefore produce overlapping features. Source conditions, such as electron-impact energy or ionization environment, influence the charge states and fragments present. A mass spectrum records the response of an ion population; neutral-mass assignments require charge-state and source information. $$ % caption: Charge-state ambiguity in a sector analyzer. At fixed mass a higher charge state bends more tightly and a lower one more gently, so charge changes the landing radius; ions of equal mass-to-charge share one radius, and a charge-state assumption is needed before a position gives a neutral mass. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,very thick,->] (-3.0,0) arc(180:0:1.4); \draw[acc!60!black,very thick,->] (-3.0,0) arc(180:0:2.4); \draw[dashed,black] (-3.0,0)--(2.7,0); \filldraw[black] (-3.0,0) circle (2pt); \node[black,left] at (-3.08,0) {entry}; \filldraw[black] (-0.2,0) circle (2pt); \filldraw[black] (1.8,0) circle (2pt); \node[below] at (-0.2,-0.08) {high z}; \node[below] at (1.8,-0.08) {low z}; \end{tikzpicture} $$ An acceleration voltage fixes kinetic energy per charge. $$ \frac{K}{z}=eV. $$ Known charge state converts voltage into kinetic energy. An unknown charge state leaves the instrument measurement as energy per charge. Combining an electrostatic-energy stage with a magnetic momentum stage can separate energy, momentum, and charge-state effects. The analysis must state which quantity is inferred: kinetic energy $K$, energy per charge $K/q$, momentum $p$, momentum per charge $p/q$, or mass-to-charge $m/q$. An analyzer can infer charge sign from bend direction only after the magnetic field orientation and detector coordinate system are calibrated. Reversing $\vec B$ should reverse the dispersed pattern. Reversing the source extraction polarity changes which charge sign enters the beamline. These reversals are practical checks against a mirrored coordinate convention or a detector readout that has been assigned the wrong direction. Energy spread broadens a spectrum even when every ion has the same $m/z$. In a magnetic sector, $r=p/(|q|B)$, so a fractional momentum spread produces the same fractional radius spread: $$ \frac{\delta r}{r} =\frac{\delta p}{p}. $$ For nonrelativistic ions at fixed mass, $\delta p/p\approx \tfrac12\delta K/K$. A source with wide extraction-energy spread therefore creates a wider detector feature. Finite selector aperture and angular acceptance add their own width. Multiple observables constrain charge state. The same ion population may be measured at two acceleration voltages, two selector settings, or with a detector capable of estimating kinetic energy. A genuine $m/z$ assignment must predict the response between settings from the instrument equations. A feature that remains at the same detector pixel while $B$, $V$, or selector ratio changes in a way inconsistent with its proposed identity is an artifact or an unmodeled beam component. ## Resolving power, peak width, and dispersion Resolution compares the separation of measured responses with their widths. Two ions with different mass-to-charge ratios can still reach a detector as one unresolved feature. The relevant coordinate is usually the detector coordinate $x$, measured along a focal plane or along a curved detector. Let $\mu=m/|q|$. Local mass dispersion is $$ \mathcal D_\mu=\frac{\d x}{d\mu}. $$ An ion group with detector width $w_x$ corresponds to an approximate mass-to-charge width $$ \Delta\mu=\frac{w_x}{|\mathcal D_\mu|}. $$ The definition applies whether $x$ happens to equal a radius, an arc length, or a pixel number after calibration. It separates two instrument properties: dispersion converts a physical change in $\mu$ into detector distance, while the image width sets how much detector distance is lost to blur. Large dispersion improves separation only while the image remains narrow. At fixed selected speed, the sector relation is linear. $$ \mu=\frac{B_a r}{v_s}, \qquad \frac{\delta\mu}{\mu} = \frac{\delta B_a}{B_a} +\frac{\delta r}{r} -\frac{\delta v_s}{v_s}. $$ Here a small fractional change in radius produces the same fractional change in $\mu$ when $B_a$ and $v_s$ are held constant. An acceleration-based analyzer has a different scaling: $$ \mu=\frac{B_a^2r^2}{2V}, \qquad \frac{\delta\mu}{\mu} = 2\frac{\delta B_a}{B_a} +2\frac{\delta r}{r} -\frac{\delta V}{V}. $$ The factor of two in the radius term matters during design. A detector blur of one part in a thousand becomes a two-part-in-a-thousand $\mu$ blur in the acceleration-based arrangement. Quoting a radius resolution without naming the analyzer mode therefore leaves the analytical resolution incomplete. Mass-spectrometry work often reports resolving power $$ R=\frac{\mu}{\Delta\mu}. $$ The width convention must accompany $R$. Full width at half maximum, standard deviation, and a specified valley criterion give different numerical values for the same pair of peaks. A report that states $R=1000$ without the peak-width convention cannot be compared reliably with another instrument. Equal-height peaks separated by one full width may look distinct on a plot, whereas a weak peak next to a strong peak can require much larger separation because the strong peak's tail raises the local background. $$ % caption: Two mass-to-charge peaks on a position-sensitive detector. Whether they are resolved depends on the centroid gap measured against the peak widths; the ideal orbit-radius difference sets the nominal separation. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(6.3,0) node[right] {detector position}; \draw[->,black] (0,-0.1)--(0,3.1) node[above] {count}; \draw[acc,very thick] plot[domain=0.3:5.4,samples=220] ({\x},{0.05+2.4*exp(-pow((\x-1.9)/0.36,2))+2.4*exp(-pow((\x-3.8)/0.36,2))}); \draw[dashed,black] (1.9,0)--(1.9,2.5); \draw[dashed,black] (3.8,0)--(3.8,2.5); \draw[black,<->] (1.9,0.62)--(3.8,0.62); \node[black,above] at (2.85,0.62) {peak gap}; \node[above] at (1.9,2.5) {peak 1}; \node[above] at (3.8,2.5) {peak 2}; \end{tikzpicture} $$ Several independent widths add to the final spot. The entrance slit has finite width, so ions begin at different transverse positions. The source emits a finite angular range, which turns a point-like entrance image into a fan of sector trajectories. A selector accepts a finite speed interval. The magnetic field differs slightly from its nominal value across the beam envelope. Collisions with residual gas and detector charge spreading can add further broadening. When the mechanisms are independent and close to Gaussian, their rms contributions combine approximately as $$ w_x^2\approx w_{\mathrm{source}}^2+ w_{\mathrm{angle}}^2+ w_{\mathrm{speed}}^2+ w_{\mathrm{field}}^2+ w_{\mathrm{det}}^2. $$ The rms combination assumes independent, near-Gaussian contributions. A clipped aperture produces shoulders and sharp edges. Multiple charge states create asymmetric composite peaks. Detector saturation can flatten a maximum. Residuals against a fitted peak model expose these departures more clearly than a single quoted width. Set slit width from resolution, count rate, detector linear range, and available acquisition time. A rectangular aperture of width $a$ transmits approximately in proportion to $a$ when incoming beam density is uniform across the aperture. Reducing $a$ sharpens the geometric image and lowers the count rate. An integrated peak area $N$ has a random counting scale of order $\sqrt N$. Source angular width requires separate treatment from source position. An entrance slit can remove positional spread while leaving a large range of directions. A collimator pair limits both, at the cost of transmission. Sector instruments with focusing geometry arrange selected angular deviations to converge near a focal surface. The focusing condition depends on sector angle and field shape. A uniform sector focuses only at the relevant focal surface. Bench alignment must verify the actual focal plane with a narrow calibration beam. Peak width also changes across a detector. A planar detector may have a constant pixel pitch while the conversion from pixel position to $\mu$ changes with position. Use the local dispersion $\mathcal D_\mu$ at each peak. A single resolving-power number commonly describes a central detector region. Calibration peaks distributed across the accepted range expose variation over the mass range. Instrument settings can favor either resolution or sensitivity. Stronger collimation, narrower selector apertures, slower scans, and lower detector gain can improve peak separability under particular limits. Each setting changes a different term in the observed width or noise model. Record the settings with the spectrum because a mass-to-charge axis omits the conditions under which a claimed separation was achieved. Source strength, detector background, and the response model set the shapes of the two curves. Very narrow apertures can place the signal below a stable background subtraction. A bright source can permit narrow slits and high resolving power. A weak transient beam may require a wider aperture and a lower resolution claim. The relevant figure of merit is the uncertainty of the requested mass-to-charge distinction during the available measurement time. ## Calibration, field integrals, and uncertainty control An analysis formula connects ideal quantities. A working beam instrument has electrode edges, finite apertures, detector pixels, supply drift, and mechanical tolerances. Calibration establishes the connection between those real components and the parameters in the formula. A spectrum should retain the calibration records needed to reconstruct its horizontal scale, its peak widths, and its stated uncertainty. The selector relation $v_s=E_s/B_s$ assumes uniform transverse fields over the entire ion path. Near the entrance and exit, both fields have fringe regions. The transverse impulse provides the more general condition. An ion that remains on the central line has $$ \Delta p_y =q\int\left(E_y-vB_z\right)\d t=0. $$ Writing $\d t=\d x/v$ along an almost horizontal path gives $$ v_{\mathrm{eff}} =\frac{\int E_y\,\d x}{\int B_z\,\d x}. $$ Uniform fields reduce this expression to $E_s/B_s$. Finite fringing changes the effective path lengths and can shift the central transmitted speed even when the plate voltage and central magnetic-field reading are correct. The shift can be measured by scanning the selector ratio with a reference beam of known speed or by comparing the selector setting against a downstream acceleration-and-sector measurement. The same issue arises in a sector analyzer. The scalar $B_a$ in $p=|q|B_ar$ represents the magnetic field sampled by the orbit. Pole-face edges, field gradients, and a displaced entrance beam change the sampled average. Mapping the usable aperture with a calibrated probe documents the spatial variation. A reference ion carried through the same aperture tests the combined effect of the map, mechanical radius scale, and fringing. That measurement includes the actual beam path as well as the map and mechanical radius scale. A position-sensitive detector requires its own calibration. Pixel number, mechanical carriage position, and physical detector distance can each be called $x$ in informal notes, although they may differ by an offset, scale factor, and weak nonlinearity. Reference ions with known $\mu$ values provide paired data $(x_i,\mu_i)$. A local calibration can be written $$ \mu(x)=c_0+c_1x+c_2x^2+\cdots. $$ Only terms supported by residuals should remain in the model. A high-order polynomial can pass through every point in a small calibration set while oscillating between points. A physically constrained sector geometry or a piecewise local calibration is usually preferable when the detector spans a broad range. Calibration residuals test the coordinate model after the reference values have been assigned. For every reference ion, calculate $$ e_i=\mu_i-\mu(x_i). $$ Plot $e_i$ against detector coordinate, nominal $m/q$, magnetic-field setting, scan time, and scan direction. A constant nonzero residual indicates an offset. A linear trend indicates an incorrect scale factor. Smooth curvature indicates that the selected calibration function is too simple over that range. Alternating residuals at successive standards can arise from a high-order interpolation that follows calibration points too closely. A time-dependent residual points toward supply drift, source-energy change, or detector motion. Compare residual size with both the required mass-to-charge separation and the scatter of repeated readings. A residual of $2\times10^{-4}$ in relative $\mu$ is harmless for a broad molecular survey and decisive when adjacent isotope features differ by a few parts in $10^4$. Reference peaks near the unknown feature carry the strongest local constraint. References at the detector ends expose global scale curvature and guard against extrapolation. Repeated measurements of one reference at fixed settings estimate short-term repeatability; measurements after a scan estimate the combined effect of drift and scan history. Keep both records because they answer different calibration questions. Calibration points must bracket the unknown peaks whenever possible. Extrapolation outside the nearest standards transfers small curvature errors into a systematic mass-scale error. Repeating the calibration before and after a scan separates a stable offset from a time-dependent drift. A temperature change in a magnet supply, electrode contamination, detector gain drift, or a moved slit can alter the response without producing an obvious discontinuity in the spectrum. The selector-sector arrangement gives a direct uncertainty model. $$ \mu=\frac{B_aB_sr}{E_s}. $$ Independent standard uncertainties combine as $$ \left(\frac{u_\mu}{\mu}\right)^2 = \left(\frac{u_{B_a}}{B_a}\right)^2 +\left(\frac{u_{B_s}}{B_s}\right)^2 +\left(\frac{u_r}{r}\right)^2 +\left(\frac{u_{E_s}}{E_s}\right)^2. $$ The equation applies to random components after known biases have been corrected. A shared current supply can correlate $B_a$ and $B_s$; then a covariance term belongs in the uncertainty calculation. A common systematic scale error also fails to average away when many spectra are collected. State whether each quoted term represents repeatability, calibration uncertainty, a bound on an uncorrected bias, or a deliberately conservative allowance. An accelerated-sector instrument has a different set of exponents. $$ \mu=\frac{B_a^2r^2}{2V}, \qquad \left(\frac{u_\mu}{\mu}\right)^2 = \left(2\frac{u_{B_a}}{B_a}\right)^2 +\left(2\frac{u_r}{r}\right)^2 +\left(\frac{u_V}{V}\right)^2. $$ The voltage term enters with a negative differential sign but a positive variance contribution. A precision voltage readout can still be misleading when the ion source loses part of the extraction voltage in a space-charge region. Reference ions accelerated under the same source conditions test the effective energy more directly than a supply display. Reversal tests identify sign and coordinate mistakes. Reversing the analyzer field reverses the bend for a beam of fixed charge. Use a documented prediction when reversing both extraction polarity and detector coordinate convention. A selector scan should have a central transmission maximum at the calibrated $E_s/B_s$ ratio. A systematic displacement between the predicted and observed maximum indicates a polarity error, a fringe correction, or a geometric offset that must be resolved before mass assignments are reported. Reproducibility requires a measurement sequence. Record zero settings, aperture widths, magnet currents, plate voltages, detector bias, source settings, scan direction, and acquisition time. Alternate unknown and reference runs when drift is plausible. Retain raw detector coordinates as well as the converted $\mu$ axis. Those records allow later recalibration if the field map, voltage correction, or reference value is revised. ## Worked selector-sector analysis > **Worked example (Mass-to-charge from a selector-sector chain).** A positive-ion > beam passes a velocity selector with $E_s=2.40\times10^4\ \mathrm{V\,m^{-1}}$ and > $B_s=0.0800\ \mathrm{T}$, then a sector analyzer with $B_a=0.2500\ \mathrm{T}$. A > detector feature sits at radius $r=0.1000\ \mathrm{m}$. Work from speed toward mass. > The selector passes only > > $$ > v_s=\frac{E_s}{B_s}=\frac{2.40\times10^4}{0.0800}=3.00\times10^5\ \mathrm{m\,s^{-1}}, > $$ > > nonrelativistic for atomic ions. The analyzer radius fixes momentum-per-charge, > > $$ > \frac{p}{|q|}=B_ar=(0.2500)(0.1000)=2.500\times10^{-2}\ \mathrm{kg\,m\,s^{-1}\,C^{-1}}, > $$ > > and dividing by the selected speed gives the mass-to-charge ratio, > > $$ > \frac{m}{|q|}=\frac{B_ar}{v_s}=\frac{B_aB_sr}{E_s}=8.33\times10^{-8}\ \mathrm{kg\,C^{-1}}. > $$ > > This is $m/|q|$, not $m$. Assigning a singly charged ion, > $|q|=e=1.602\times10^{-19}\ \mathrm{C}$, gives > > $$ > m=(8.33\times10^{-8})\,e=1.34\times10^{-26}\ \mathrm{kg}\approx8.04\ \mathrm{u}. > $$ > > A doubly charged ion at the same spot would read $16.1\ \mathrm{u}$; the sector alone > cannot choose. Charge state must come from the source, isotope structure, or a second > measurement. Uncertainty enters before the charge-state assignment. Suppose each magnetic field calibration and the selector electric-field calibration has relative standard uncertainty $0.050\%$, while the fitted radius has uncertainty $0.150\%$. Treating these terms as independent gives $$ \frac{u_\mu}{\mu} =\sqrt{ (0.00050)^2+ (0.00050)^2+ (0.00150)^2+ (0.00050)^2} =0.00173. $$ The standard uncertainty in the measured ratio is therefore $$ u_\mu =(8.33\times10^{-8})(0.00173) =1.44\times10^{-10}\ \mathrm{kg\,C^{-1}}. $$ The radius term dominates this example. Reducing detector-position uncertainty by half would improve the combined result more than reducing one of the already small field-calibration terms by half. The actual error budget determines the priority. Instrument changes should target the largest validated contribution. Peak resolution can be evaluated with the same geometry. A neighboring feature separated by $0.200\%$ in $\mu$ has, under fixed selected speed, an ideal radial separation $$ \Delta r =r\frac{\Delta\mu}{\mu} =(0.1000\ \mathrm m)(0.00200) =0.200\ \mathrm{mm}. $$ An observed full width of $0.150\ \mathrm{mm}$ may allow two equal-height features to be distinguished under a stated width convention. A width of $0.300\ \mathrm{mm}$ produces substantial overlap. The result also depends on intensity ratio and background. A numerical resolution claim must state both the peak separation rule and the calculated value of $\Delta r$. An acceleration-based cross-check can use the same target radius. Setting $V=3.75\ \mathrm{kV}$ gives $$ \frac{m}{|q|} =\frac{B_a^2r^2}{2V} =\frac{(0.2500\ \mathrm T)^2(0.1000\ \mathrm m)^2} {2(3.75\times10^3\ \mathrm V)} =8.33\times10^{-8}\ \mathrm{kg\,C^{-1}}. $$ Agreement constrains selector calibration, analyzer field scale, radius calibration, and extraction-energy interpretation together. A selector fringe correction, a voltage loss near the source, an incorrect effective radius, or a charge-state mixture can produce the same mismatch. Isolate the cause by changing one control at a time and comparing the observed shift with the relevant dependence on $E_s$, $B_s$, $B_a$, $V$, or $r$. ## Measurement sequence, data reduction, and reporting Beam analysis begins with a stable reference state. Set the source extraction voltage, source current, aperture positions, selector ratio, analyzer current, detector bias, and vacuum condition before interpreting a peak position. A spectrum recorded while any of these settings drifts is a mixture of instrument states. The raw data should retain a time coordinate, the unconverted detector coordinate, and every scanned control value. The scan variable determines the form of the calibration. A fixed detector array can collect an entire dispersed range at one magnet current. A scanning instrument may move a detector, vary the analyzer magnetic field, or vary the acceleration voltage. Magnetic-field scans follow the relation $$ \mu\propto B_a \quad\text{at fixed }r\text{ and }v_s, $$ for a selector-sector layout, whereas an acceleration-sector layout at fixed radius has $$ \mu\propto \frac{B_a^2}{V}. $$ Use the control setting recorded for each datum in the axis conversion. A field scan also requires a current-to-field calibration; direct proportionality between coil current and $B_a$ can fail near magnetic saturation or after an incomplete hysteresis cycle. Reference measurements should bracket an unknown run in time. A reference peak before the run establishes the initial coordinate scale. A second reference after the run detects drift. Interleaved references are needed when the target precision approaches the observed drift rate. If the two references disagree, report the disagreement and interpolate only when the time dependence is supported by additional reference points. A single before-and-after pair detects a net change but cannot establish that drift was linear. Peak reduction begins with separate background, detector-response, and fit checks. | check | control measurement | invalid result signature | |---|---|---| | background | source blocked or ion-excluding setting | drifting baseline or structured residual | | detector linearity | known attenuation or source-current change | peak height or area no longer proportional | | peak model | baseline plus neighboring features | patterned centroid or width residual | | reproducibility | repeated complete scans | between-scan shift beyond fit uncertainty | - **Background.** Record a trace with the source blocked or with a setting that prevents ions from reaching the detector. Dark counts, electronic pickup, residual-gas ions, and scattered particles can all contribute. Check baseline stability before subtraction; model a changing background explicitly rather than treating it as a constant offset. - **Detector response.** Gain sets the vertical scale and can change apparent peak shape. A counting detector can lose close arrivals through dead time; an analogue detector can saturate near a tall peak. Both suppress strong features more than weak ones and can corrupt isotope ratios or charge-state fractions despite a correct horizontal mass scale. Keep height or area comparisons inside the validated detector range. - **Peak fit and repetition.** Extract centroid, width, area, and uncertainty from a model that includes baseline, peak shape, and overlapping neighbors. An asymmetric tail, flat top, or shoulder requires a different model or a narrower claim. Partially resolved groups require simultaneous fitting; a fixed integration window transfers area between groups as their positions change. Separate within-scan fit uncertainty from between-scan scatter caused by source instability, magnet drift, mechanical motion, or changing background. Retain the following in the reduction record. - **Raw trace**: detector coordinate, counts, time, and the complete control sequence before conversion to a mass-to-charge axis. - **Reference set**: assigned reference values, their detector positions, calibration residuals, and the time at which each reference was measured. - **Peak model**: baseline form, peak-shape function, fitting interval, centroid, width convention, integrated area, and residual plot. - **Instrument configuration**: slit settings, selector voltage and magnetic field, analyzer magnetic field, acceleration voltage, source condition, and detector bias. - **Uncertainty statement**: random repeatability, calibration terms, any common systematic scale term, and the stated confidence convention. Detector position and magnetic-field scan direction should be recorded. A magnet can follow different field-current paths while increasing and decreasing current because of hysteresis. The consequence is a peak position that shifts with scan direction even when the current readout returns to the same number. Precycling the magnet through a documented range and approaching each measurement from the same direction reduces this ambiguity. Alternating upward and downward scans exposes a residual direction dependence. Instrument artifacts have recognizable parameter dependences. A genuine magnetic-sector peak moves with $B_a$ according to the analyzer relation. A feature caused by detector electronics may remain at one detector channel while the expected mass coordinate shifts. A source-space-charge effect often changes with beam current and extraction voltage. A residual-gas feature may track chamber pressure or source temperature. Stray electric fields can displace the beam in a way that reverses with charge sign. Each diagnostic uses an explicit control change and a predicted response. Identity evidence comes from the observed response to that control change. Charge-state assignments require an extra statement beyond a measured $m/z$. An observed $m/z=14$ peak can represent a singly charged approximately-14-unit ion, a doubly charged approximately-28-unit ion, or a fragment whose neutral precursor was heavier. Isotopic spacing, source chemistry, known extraction polarity, and energy-per-charge measurements can reduce the alternatives. Ambiguous charge-state evidence requires the peak to be reported as $m/z$, with the alternative particle assignments listed. The final result should state the observable and the conditions. A concise technical report gives the measured $m/|q|$, its uncertainty and width convention, the calibration standards, the selector and analyzer settings, the detector-coordinate model, the charge-state evidence, and unresolved alternatives. Include raw or minimally processed data when the conclusion depends on a line-shape choice, background subtraction, or a drift correction. That level of record keeps later recalibration possible and separates a measured trajectory property from an inferred particle identity. ## Model limits and independent closure checks The preceding analyzer equations use nonrelativistic momentum, $p=mv$ and kinetic energy $K=\tfrac12mv^2$. Their accuracy depends on the ratio of beam speed to the speed of light. A velocity selector retains the transverse balance $E=vB$ for perpendicular fields, because the electric and magnetic force magnitudes still balance at the selected speed. The conversion from radius and speed into mass changes once $v$ is large enough that relativistic momentum differs appreciably from $mv$: $$ p=\gamma mv, \qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}. $$ The sector measurement remains a measurement of momentum per charge. $$ \frac{p}{|q|}=B_ar, $$ Relativistic mass inference uses $\gamma mv$. Acceleration through voltage obeys $$ K=(\gamma-1)mc^2=|q|V $$ when the extraction energy is dominated by the applied voltage. A nonrelativistic reduction at high speed shifts the mass scale systematically. Independent calibration is required because repeatability cannot expose this common scale error. The nonrelativistic source-energy relation provides an independent comparison. If an ion begins with negligible kinetic energy and gains $|q|V$, then $$ \frac12mv_s^2=|q|V, \qquad \frac{m}{|q|}=\frac{2V}{v_s^2}. $$ The selector measures $v_s=E_s/B_s$. The sector measures $m/|q|=B_ar/v_s$. The source-voltage relation gives a third estimate: $$ \frac{m}{|q|} =\frac{2V B_s^2}{E_s^2}. $$ All three estimates should agree within their stated uncertainty when the source energy, selector ratio, and analyzer geometry are described correctly. The comparison is especially sensitive to an extraction-voltage loss or an incorrect selector effective-field integral. It also identifies a source whose ions leave with substantial initial energy, since the simple $|q|V$ energy gain no longer accounts for the selected speed. The source-energy relation also has a charge-state consequence. A source at fixed extraction voltage gives a larger kinetic energy to an ion with larger $|q|$. At fixed mass, a multiply charged ion therefore enters the selector with a higher speed. A separate source or selector setting can compensate. The selector removes most of that speed variation by transmitting a narrow band. Its transmitted intensities can still be strongly charge-state dependent. Source and transmission corrections are required before peak height is used as a neutral-abundance measurement. Neutral particles travel undeflected through ideal crossed and magnetic fields. A neutral contribution can reach a detector through a direct line of sight, scattering, secondary ionization, or a detector response unrelated to the analyzed trajectory. Beam stops and apertures should block the straight neutral path before the detector. A persistent straight-through signal during magnetic-field reversal is a practical indication of such a contribution. Residual gas affects both trajectory and intensity. A collision can scatter an ion out of the accepted angle range, change its charge state, produce a fragment, or create a broad background. Increasing pressure generally lowers the unscattered beam current and increases these nonideal components. The effect is strongest for long flight paths and wide energy distributions. Vacuum pressure belongs in the instrument record when weak peaks or charge-state fractions are interpreted. Space charge limits the single-particle picture at high beam current. A dense positive ion beam has its own electric field. The resulting transverse defocusing changes the source emittance, selector acceptance, and detector spot width. A current scan at unchanged nominal fields distinguishes this effect from a static geometric error: a space-charge contribution changes with beam current, whereas a fixed mechanical offset remains approximately constant. Reduced source current, wider source extraction spacing, or additional focusing electrodes can reduce the beam-density contribution. Each adjustment requires a fresh calibration of transmission and mass scale. Assign a particle mass after the checks agree. Verify that field and voltage units give the expected dimensions, reference peaks occur at calibrated coordinates, field reversal yields the predicted bend reversal, a control scan follows the correct dependence, and source-energy, selector, and sector estimates agree where their assumptions apply. Report a remaining discrepancy as an unresolved systematic effect. An accurate mass scale requires these checks in addition to a narrow spectrum.