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The citation panel credits Jay Abramson as author. The preface identifies him as lead author and senior contributing author, and lists a contributing team without assigning individual chapters. OpenStax publishes and hosts the book.

Whole-book publication: 21 December 2021, as given by the citation panel. The ©2026 footer is distinct from that publication date; no section revision dates are established.

Publisher: openstax.org/details/books/precalculus-2e

Author-role metadata: About the Authors, Senior Contributing Author and Contributing Authors; citation information

Online text: crypto.stanford.edu/pbc/notes/numbertheory/qr.html

By Ben Lynn. Stanford hosts the author's PBC Notes; it is not an institutional author. The inspected page is undated.

Cited passages: “Quadratic Residues”, opening definition and Euler criterion discussion; “The Legendre Symbol”, three values, odd-prime restriction, multiplication and square factors. The opening square-residue convention includes zero.

Online text: kconrad.math.uconn.edu/blurbs/ugradnumthy/QuadraticResiduePatterns.pdf

By Keith Conrad. These standalone notes are hosted on the author's University of Connecticut website. No document date is established.

Cited passages: printed p. 1, title/byline and §1 opening paragraph: nonzero convention and equal counts of squares and nonsquares; printed p. 2, §2 opening notation for the Legendre symbol.

Annealing uses controlled heating and cooling to change a solid's internal structure and properties. Its outcome depends on the temperature, duration and cooling history.

Sputtering ejects surface chemical atoms from a solid through bombardment by energetic incident particles, often ions.

Sintering joins particles into a coherent solid by heating, sometimes with applied pressure, without completely melting the material. It is used to consolidate powders of ceramics and metals.

A ceramic is an inorganic, nonmetallic solid, often formed by heating and cooling. Ceramics may contain several chemical compounds and need not have a single crystal structure.

Tensile testing measures the response of a specimen to an axial pulling force. A tensile modulus experiment uses the initial proportional response to infer Young's modulus.

For a modern chosen example, the National Institute of Standards and Technology lists neutral magnesium atomic masses $23.985041697\mathrm{u}$, $24.985836976\mathrm{u}$, and $25.982592968\mathrm{u}$, with reference isotopic abundances $0.7899$, $0.1000$, and $0.1101$. They describe the isotopes with mass numbers 24, 25, and 26, respectively.^[ref] These modern inputs are not a reconstruction of historical observations.

Use the dimensionless mass-to-charge ratio coordinate $x=(m/\mathrm{u})/z$. For the illustration only, approximate ion masses by the neutral atomic masses. The singly charged positions are then near $23.98504$, $24.98584$, and $25.98259$; the doubly charged companions are near $11.99252$, $12.49292$, and $12.99130$. The neutral-to-ion correction already given above is essential at higher accuracy. A peak near $x=12$ alone need not identify a chemical element: another ion with a different electric charge can occupy the same mass-to-charge ratio interval.

For the narrow singly charged peak at $x\approx 24.98584$, the chosen full width at half maximum $\Delta x=0.12$ gives mass resolving power $R\approx 208.2$; widening to $\Delta x=1.20$ gives $R\approx 20.82$. Adjacent isotope spacings here are approximately one unit in $x$. Peaks are visibly separated in the first chosen profile and overlap strongly in the second. A small uncertainty in a fitted peak centre would not by itself make the overlapping components resolved.

Even integrated peak areas need a response model. For isotope fraction $f_i$, let ${\eta }_{i,z}$ collect the probability of ionization into electric charge state $z$, transmission through the analyzer, and the detection response. In a linear instrument with overall exposure factor $C$, the component area isEqual ${\eta }_{i,z}$ at a fixed $z$ makes normalized areas equal the fractions; otherwise it does not. In the chosen example, doubling only the mass-25 response changes the normalized areas to $0.71809$, $0.18182$, and $0.10009$, although the neutral sample is unchanged. Photographic darkening introduces another possible nonlinearity: neither line darkness nor peak height is a universal measure of neutral isotopic abundance. Calibration of position and calibration of intensity answer different questions. Matched standards, a known electric charge assignment, and a response correction make isotope analysis useful; a visually impressive set of peaks alone does not supply those controls.

Reuse the velocity selector analysis in the Thomson charge-to-mass experiment, but now send positive ions with $q=ze$ into a separate magnetic analyzer. The selector fields $E_s$ and $B_s$ give $v_0=E_s/B_s$. If the analyzer field $B_a$ is perpendicular to the selected velocity, its gyroradius givesfor a semicircular return to the entrance plane. The displayed impact separation $D$ is a diameter. This formula is linear in $r$ at fixed selected speed; the fixed-electric potential difference model above instead gives $m/q\propto r^2$. They impose different experimental conditions.

For a chosen example, take $E_s=2.00\times 10^{4}V/m$, $B_s=0.200\mathrm{T}$, and $B_a=0.500\mathrm{T}$. Thus $v_0=1.00\times 10^{5}m/s$. Approximating an ion mass by $24\mathrm{u}$ and setting $q=e$ gives $r\approx 49.75\mathrm{mm}$ and $D\approx 99.50\mathrm{mm}$. Here $\mathrm{u}$ denotes the unified atomic mass unit. These are stipulated teaching inputs, not a reported Aston measurement or an exact magnesium atomic mass.

A finite accepted speed range broadens the radius distribution. At fixed $m/q$ and $B_a$, $\delta r/r=\delta v/v$. At fixed electric charge and mass, $K=m{v}^2/2$ gives $\delta v/v\simeq \delta K/(2K)$ for a small kinetic energy spread. A one-percent kinetic energy spread therefore produces roughly a half-percent radius spread before other broadening is included. Its width convention must be specified before converting that statement into a mass resolving power.

Accelerator mass spectrometry uses ion acceleration and particle identification to count rare isotopes relative to abundant ones. For radiocarbon dating, it measures carbon-14 directly rather than waiting to count its beta decays.

A scientific collaboration, not an individual person.

An automatic gas-lighting system associated with Gustaf Dalén. The sun valve decides whether daylight permits the main supply; a separate flashing mechanism determines the repeated light pattern. Keeping those functions distinct is useful when identifying what an apparatus actually controls.

Official award records: www.nobelprize.org/prizes/lists/all-nobel-prizes-in-physics/