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A grating spectrometer places an illuminated narrow slit at the front focal plane of a collimator. The resulting parallel beam reaches the diffraction grating at known incidence, here taken normal. A rotatable telescope focuses each outgoing diffracted parallel beam onto its crosshair or detector. Align the slit with the grating rulings, focus the collimator at infinity, and illuminate many grooves across a suitable aperture. A narrow slit supplies sufficient spatial coherence and prevents slit width from blurring the angular separation; the telescope must also resolve the two images.

The groove spacing is $d=(2000\times 100{)}^{-1}\mathrm{m}=5.00\mu \mathrm{m}$. The grating equation at normal incidence is $d\sin {\theta }_m=m\lambda$. For diffraction order $m=2$,ThereforeThe ideal resolving power of a diffraction grating is $|m|N$, so the Rayleigh criterion for this doublet requires about $N\ge 145$ illuminated grooves, spanning roughly $0.73\mathrm{mm}$. This is an ideal lower bound; finite slit width, aberrations and detector resolution can require a larger usable aperture.

After injection, the radiopharmaceutical circulates and accumulates according to its biological targeting and clearance. Technetium-99m undergoes an isomeric nuclear transition, producing approximately $140\mathrm{keV}$ gamma rays; its roughly six-hour half-life permits useful counting with limited persistence. Some photons undergo absorption or Compton scattering in the patient. A lead collimator in the gamma camera accepts a narrow range of directions so the accepted event retains positional information.

An accepted gamma ray deposits energy, commonly in a NaI(Tl) scintillator, by photoelectric effect or Compton scattering. The resulting visible particle scintillation light is collected by photomultipliers. Their relative pulse sizes determine an event position and their summed pulse estimates its energy. A pulse-height analyser accepts an energy window around the photopeak to reject many scattered events. Position and uniformity corrections are applied; accepted events increment a two-dimensional pixel histogram. After acquisition the computer displays this count map as a planar projection of tracer activity. Finite counting statistics, attenuation, scattering and the collimator limit the image's quantitative accuracy and spatial resolution.^[ref]

Single-photon emission computed tomography detects individual gamma rays from a tracer. Mechanically collimated gamma cameras acquire projections from multiple angles for tomography. The collimator discards most emitted photons, creating a strong sensitivity–spatial resolution trade-off.

Positron emission tomography uses a Positron-emitting tracer. The emitted Positron travels a short distance before electron-positron annihilation produces an approximately back-to-back photon pair. Coincident detection in a detector ring defines a line of response, giving electronic collimation and greater detection efficiency; time-of-flight PET further constrains position along that line. The positron range, non-collinearity, detector size, scatter and accidental coincidences limit spatial resolution and quantification. In both hybrids, computed tomography supplies anatomy and an attenuation correction map, with attention to registration and respiratory motion.^[ref]

A prism minimum-deviation index measurement infers a transparent sample's relative refractive index from two angles: its known apex angle $A$ and the smallest deviation ${\delta }_{\min }$ of a selected spectral line. A spectrometer combines a source and slit, a collimator, a rotating optical prism table, and a viewing optical telescope with an angular readout. The collimator supplies a nearly parallel beam; the optical telescope locates its emergent direction. This turns an otherwise difficult measurement of the phase velocity of light into a measurement of angles.

For the symmetric path through a uniform optical prism, with the same external medium at both faces and positive refractive indices,The angles $i,e,r_1,r_2$ are measured from face normals. This is the minimum deviation relation, not a formula for an arbitrary prism orientation. An absolute refractive index additionally requires $n_{\mathrm{ext}}$.

For example, stipulate $A=60^{\circ }$ and $n_{\mathrm{rel}}=3/2$. Then $i=e=\arcsin (3/4)\simeq 48.5904^{\circ }$ and ${\delta }_{\min }\simeq 37.1808^{\circ }$. Substitution recovers $n_{\mathrm{rel}}=1.5$. These synthetic values illustrate the geometry; they are not readings for the prism in the laboratory protocol below.

A Michigan State University teaching experiment uses a mercury lamp, slit, collimator, prism table and optical telescope, with a subsidiary angular scale read to one arcminute. First focus the optical telescope on a distant object; with the prism removed, align its crosshair to a narrow vertical slit image, focus the collimator, and establish the direct-beam zero. Insert and centre the prism. Select the green mercury spectral line, tabulated at $546.1\mathrm{nm}$, then rotate the prism while tracking its image with the optical telescope. Locate the turning point where continued prism rotation reverses the image motion, and read the deviation. Re-establish the minimum separately for each other spectral line. The protocol assumes $A=60^{\circ }$; that assumed apex angle is not an independently measured result. ^[ref]

For an uncertainty-aware implementation, repeat the turning-point search from both prism-rotation directions, recheck the direct-beam reference, and record individual readings before averaging. Independently determine $A$ or include its calibration uncertainty; record the chosen spectral line, sample identity, temperature and surrounding-air conditions. These are proposed controls deduced from the inference, not claimed additional instructions or results of that manual. Finite slit width, unresolved spectral lines, imperfect collimation, table tilt, angular calibration and sample nonuniformity can limit the inferred refractive index. One arcminute readout resolution alone does not establish one arcminute total measurement uncertainty. A single selected wavelength gives one index, not a measured optical dispersion curve.

A short geometric derivation uses $r_1+r_2=A$ and $\delta =i+e-A$. Reversibility exchanges entrance and exit. At the symmetric stationary path, $i=e$ and $r_1=r_2$; applying Snell's law gives the displayed index relation. The plotted calculation checks that this stationary point is a minimum on the chosen transmitting branch. To assess local sensitivity, differentiate the inferred index using angles in radians:This first-order relation exposes both angular contributions; it is not an instruction to add signed systematic errors as independent random errors. In the synthetic example, a deviation error of $+1$ arcminute at fixed $A$ changes the inferred relative refractive index by approximately $+1.92\times 10^{-4}$.

In 1913 Georges Sagnac mounted a lamp, collimator, beam splitter, four mirrors and photographic plate on a horizontal rotating platform. The mirrors defined a closed polygon; the two beams traversed it in opposite senses. A slight beam splitter adjustment produced visible interference fringes. A motor brought the platform to steady rotation, and switching the lamp exposed the plate. Comparing opposite senses of rotation revealed a reversing displacement. Changing the beam splitter adjustment supplied a further check against optical deformation. The report compared measured and calculated displacements. Georges Sagnac interpreted this as evidence for a luminiferous aether; the relativistic derivation below reproduces the delay without that hypothesis.^[ref]

The readout measures a relative wave phase, not two stopwatch readings separated by an optical transit time. Reversal isolates a contribution odd in angular velocity: if a signed interference fringe position is $q(\Omega )=q_0+k\Omega$ to leading order, then $[q(\Omega )-q(-\Omega )]/2=k\Omega$ cancels the fixed offset $q_0$. Changes of alignment or temperature correlated with reversal can survive this subtraction. Thus the experimental inference concerns a reproducible rotating-loop phase difference, with controls for disturbances; it is not by itself a unique test of an underlying propagation hypothesis.

A slit of width $x$ at the focus of a collimator subtends $x/f_{\mathrm{coll}}$. Its projected width is $x{f}_{\mathrm{cam}}\cos \alpha /(f_{\mathrm{coll}}\cos \beta )$. Dividing this width by the grating dispersion gives $\Delta \lambda =xd\cos \alpha /(m{f}_{\mathrm{coll}})$. Thus the spectral resolving power is $R=m\lambda f_{\mathrm{coll}}/(xd\cos \alpha )$. Omitting the anamorphic factor gives a denominator $\cos \beta$ instead; these agree in the Littrow configuration.

In 1913 Johannes Stark reported that an applied electric field resolved the hydrogen $H_{\beta }$ and $H_{\gamma }$ spectral lines into five components at his instrument's spectral resolving power, viewed perpendicular to the electric field. Three inner components had polarization of an electromagnetic wave perpendicular to the electric field; two outer components had it parallel.^[ref]

In his lecture of 3 June 1920 for the 1919 prize, Johannes Stark describes canal rays emerging through a perforated cathode into the electric field between it and a supplementary electrode. A spectrograph recorded the emitted light. His earlier Doppler effect comparison used a collimator perpendicular to the beam and then along approaching rays. Later beam/electric field reversal changed component intensities.^[ref]

The historical account distinguishes observation perpendicular to the electric field, with five resolved components, from observation parallel to it, with three. These are viewing directions, distinct from the direction of the polarization of an electromagnetic wave.^[ref]

The experimental distinctions matter. Changing the viewing direction tests a Doppler effect contribution; changing the applied electric field tests the Stark effect; analyzing polarization of an electromagnetic wave separates components that may overlap in an unpolarized record. A comparison with the unperturbed spectral line fixes the origin of a displacement. These are different controls, and the earlier Doppler effect comparison should not be described as a field-off control of the later discovery. A spectrograph's finite spectral resolving power can merge several components into one. In particular, a count of visible spectral lines does not by itself count all energy eigenstates.

For a modern idealization of the gap, an electric potential difference $\Delta V$ over separation $d$ gives $|\mathcal{E}|\simeq |\Delta V|/d$ only if edge effects and the charge density in the gas are negligible. This is a calibration model, not a historical value assigned to Stark's apparatus. Similarly, a canal ray beam need not be a pure beam of neutral excited hydrogen atoms; the quantum mechanics model below isolates one neutral hydrogen atom from the source's preparation and collision processes.

In 1922, Otto Stern and Walther Gerlach used an electrically heated oven to emit neutral silver chemical atoms. Two platinum apertures, a pinhole and a slit $0.03$–$0.04\mathrm{mm}$ wide, supplied collimation. The beam passed along a $3.5\mathrm{cm}$ knife-edge pole in a nonuniform magnetic field. At its end, a glass plate collected the condensed silver. Continuous pumping maintained vacuum; the thin deposit required development even after eight hours.^[ref]

The field-off image established the undeflected slit width. A comparison put field-on and field-off deposits on the same plate. Microscopy revealed two deflected bands, no detectable central component, and broadening attributed to the velocity distribution. Alignment and distance from the knife-edge affected the pattern; precise magnetic field gradient measurements were still pending.^[ref]

The apertures act as a collimator: they constrain the initial transverse positions and velocities, so a magnetic displacement can be distinguished from an initially broad beam. Vacuum reduces collisions that would destroy this preparation. Deposition integrates arrivals over the exposure; a darkened band is not a time-resolved record of individual chemical atoms. The field-off comparison tests whether the magnet changes the spatial distribution, while alignment checks address a geometric source of apparent asymmetry. Finite sensitivity qualifies the absence of a central deposit: it is an experimental upper limit, not a proof that exactly zero atoms landed there.

For a narrow beam of monoenergetic photons crossing homogeneous material of thickness $x$, the uncollided wave intensity obeys $I(x)=I_0{e}^{-\mu x}$. Here $\mu$ is the linear attenuation coefficient, counting removal from the incident beam by photon absorption and scattering of light. It depends on photon energy and material; it is not universally independent of energy.^[ref]

An original derivation makes the model explicit. If each thin layer removes fraction $\mu dx$ of the surviving beam, with the same $\mu$ throughout, thenThus equal added thicknesses multiply transmission by the same factor. The ordinary differential equation describes uncollided beam survival. To test it with a detector, keep source output, distance, selected photon energy band and detector response fixed. Use a collimator and suitable detector acceptance to exclude scattering of light into the detector. Subtract background, measure an empty-path reference, and keep the detector in its linear operating range. A changing source distance adds geometric dilution and is not an attenuation measurement at fixed geometry.

For a stipulated example, take $\mu =0.40\mathrm{m}{\mathrm{m}}^{-1}$ and $x=2.5\mathrm{mm}$. Transmission is $e^{-1}\simeq 0.368$. If the empty-path detector rate is $1010{\mathrm{s}}^{-1}$, including background $b=10{\mathrm{s}}^{-1}$, the model predicts sample rate $10+1000e^{-1}\simeq 378{\mathrm{s}}^{-1}$. Recover transmission as $(R-b)/(R_0-b)$; the raw ratio $R/R_0\simeq 0.374$ is biased. These chosen values are a calculation, not measurements of a particular material or a historical apparatus.

An X-ray tube accelerates electrons from a negative cathode to a positive target anode. A heated cathode supplies electrons by thermionic emission; the evacuated envelope limits collisions before the target. This differs from the gas-discharge tube in the Röntgen X-ray discovery experiment.^[ref]

In bremsstrahlung, an electron changes velocity in the target's electric fields, emitting a photon while losing kinetic energy. Different encounters allow a continuous range of transferred energies. This is distinct from the discrete electronic transitions producing Characteristic X-rays.

Changing the accelerating voltage changes the available kinetic energy per electron and the X-ray tube cutoff. Changing the number of arriving electrons changes the number of opportunities for photon emission. These are distinct controls: a larger electric current does not by itself give each electron more energy. For an original idealized budget, $V=50\mathrm{kV}$ and tube electric current $i=0.20\mathrm{mA}$ give electrical power $Vi=10\mathrm{W}$ and arrival rate $i/e\simeq 1.25\times 10^{15}{\mathrm{s}}^{-1}$, where $e$ is the elementary charge. This is input power, not emitted X-ray power; obtaining the latter requires a conversion efficiency.