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Laboratory chapter in Introductory Physics Experiments, Physics 252, version 4.0 (Michigan State University). Version 4.0 and date 1/7/2002 are printed together; the day/month order is left uninterpreted. This is a version date, not an established first publication date. Michigan State University hosts the manual; no individual chapter byline or institutional authorship credit is inferred.

Official chapter PDF: web.pa.msu.edu/courses/2002summer/PHY252/Prism.pdf

Cited passages: pp. 1–3, objectives, minimum-deviation relation and symmetry; pp. 5–8, apparatus, alignment, vernier angle readout and mercury-line turning point; p. 9, supplied mercury wavelengths. This is a teaching procedure, not a report of newly measured data.

For nondegenerate bands separated by $E_g$, the intrinsic carrier concentration is $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2k_{B}T)}$, using the effective density of states $N_c,N_v$. For fixed parabolic band effective masses and fixed gap, $n_i\propto T^{3/2}{e}^{-E_g/(2k_{B}T)}$.

For nondegenerate conduction bands and valence bands, charge neutrality gives $N_c{e}^{-(E_c-{\mu }_i)/(k_{B}T)}=N_v{e}^{-({\mu }_i-E_v)/(k_{B}T)}$. Taking logarithms yieldsFor single spin-degenerate parabolic extrema this becomes $(E_c+E_v)/2+(3k_{B}T/4)\ln (m_h^{\ast }/m_e^{\ast })$. Valley multiplicities or additional hole bands belong in $N_c,N_v$ and change the logarithm. The intrinsic chemical potential lies exactly at midgap only when the two effective density of states values agree.

Section of Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis. No separate chapter author or section publication date is credited. The parent edition was published 6 May 2020.

Online text: openstax.org/books/intermediate-algebra-2e/pages/9-2-solve-quadratic-equations-by-completing-the-square

Cited passages: §9.2, Binomial Squares Pattern; Solve Quadratic Equations of the Form ax²+bx+c=0; Citation information.

Chapter of Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. Whole-work author credits apply; no separate chapter author byline is established. No chapter publication or revision date is established; the URL path does not establish a date or edition.

Online text: interactivetextbooks.tudelft.nl/interactive-optics/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html

Cited passages: Scalar Diffraction Optics: Fresnel Approximation, equations (381)–(385), following paraxial equation remark; Fresnel and Fraunhofer Approximations, equations (386)–(392): omitted aperture quadratic phase, full-diameter Fresnel number; About the authors: biographies and final whole-book author credit.

Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.

Online text: textbooks.math.gatech.edu/ila/one-to-one-onto.html

Cited passages: §3.2.1, One-to-one matrix transformations theorem and Wide matrices do not have one-to-one transformations; §3.2.2, Onto matrix transformations theorem and Tall matrices do not have onto transformations.

Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.

Online text: textbooks.math.gatech.edu/ila/row-reduction.html

Cited passages: §1.2.1, Augmented Matrices and Row Operations: preservation of solutions; §1.2.2, Echelon Forms: leading entries and reduced form; §1.2.3, definitions of pivot position and pivot column.

Online text: textbooks.math.gatech.edu/ila/projections.html

Section of Interactive Linear Algebra by Dan Margalit and others, by Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted. The parent record's 2019 year dates the front matter; the colophon copyright is 2017.

Cited passages: §6.3.1, Orthogonal decomposition theorem and Closest vector and distance; §6.3.2, Properties of Orthogonal Projections.

A scientist or clinician studying the immune system and its disorders. An immunologist connects molecular recognition and cellular responses with protection, regulation or immune-mediated injury.

An interacting system of circulating and surface-associated immune proteins that helps mark targets, recruit responses and damage susceptible membranes. Its activation is regulated to limit host injury, and some pathways connect antibody recognition with downstream immune effects.

Section 5.6 of Hyper-TVT by ETH Zurich. No personal chapter author is credited. ETH Zurich’s Institute of Process Engineering issued the teaching site; the ETH Web Archive hosts the preserved text. No publication or revision date is established.

Online text: www.hyper-tvt.ethz.ch/distillation-efficiency.html

Cited section: 5.6 Efficiency, opening paragraphs: equilibrium plates versus actual plates; Overall efficiency paragraph: hydrodynamics and mass transfer affect efficiency; Lower efficiency increases actual stage count; Site imprint, Publisher and Webmaster fields.

The Lyman series consists of hydrogen transitions terminating in the $n=1$ ground state. Its emission wavelengths obey $1/\lambda =R_H(1-1/n_i^2)$ for $n_i=2,3,\dots$. The $2p\to 1s$ electric-dipole line is the Lyman-alpha line, and the accumulation point is the Lyman limit.

These larger energy differences put the series in the ultraviolet rather than the visible Balmer range. Observing it requires suitable ultraviolet optics and a detector whose response includes those wavelengths; ordinary visible-light transmission is not sufficient.

As the upper level approaches the ionization threshold, the lines crowd together. Above that threshold, absorption from the ground state can ionize the atom instead of producing a higher discrete bound level. The Rydberg formula describes the line positions, while populations, radiation transport and instrumental resolution determine which lines are distinguishable.

The Balmer series consists of hydrogen transitions terminating at principal quantum number $n=2$. Its emission wavelengths obey $1/\lambda =R_H(1/4-1/n_i^2)$ for $n_i=3,4,\dots$, a special case of the Rydberg formula.

The first transitions fall in the visible part of the electromagnetic spectrum, while the high-level lines converge toward a shorter-wavelength limit with ${\lambda }_{\lim }=4/R_H$. A hydrogen discharge observed through a spectroscope shows separated lines rather than a smooth continuum because the atomic energy differences are discrete.

The same level separations can produce absorption when atoms initially occupy the lower state. Temperature, excitation and the number of atoms in that state affect line strength; a wavelength formula alone does not predict the observed relative intensities.