= Prism minimum-deviation index measurement
{title2=$n_{\rm rel}=\sin[(A+\delta_{\min})/2]/\sin(A/2)$}
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>,
$$
r_1=r_2=\frac A2,\qquad i=e=\frac{A+\delta_{\min}}2,\qquad n_{\rm rel}=\frac{n_{\rm prism}}{n_{\rm ext}}=\frac{\sin[(A+\delta_{\min})/2]}{\sin(A/2)}.
$$
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 $n_{\rm ext}$.
\Image[/refractive-index-prism-minimum-deviation.png]
{title=<Prism minimum-deviation index measurement>: symmetric ray geometry and an independently calculated deviation minimum}
{description=The original model chooses $A=60^\circ$ and $n_{\rm rel}=3/2$ at one fixed <wavelength>. The left panel marks face normals, equal internal <angles>, apex <angle> and deviation from the undeflected incident direction. The right panel calculates deviation versus incidence from two applications of <Snell's law>; the marked minimum is the only orientation used for the index inference. No points are experimental observations.}
For example, stipulate $A=60^\circ$ and $n_{\rm rel}=3/2$. Then $i=e=\arcsin(3/4)\simeq48.5904^\circ$ and $\delta_{\min}\simeq37.1808^\circ$. Substitution recovers $n_{\rm rel}=1.5$. 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. <introductory-physics-experiments-physics-252-version-4-0-michigan-state-university/The Prism Spectrometer>[introductory-physics-experiments-physics-252-version-4-0-michigan-state-university; Michigan State University, The Prism Spectrometer, Physics 252 v4.0: pp. 2, 5–9; apparatus, vernier readout, alignment, turning point and wavelength table]{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:
$$
\frac{dn_{\rm rel}}{n_{\rm rel}}=\frac12\cot\!\left(\frac{A+\delta_{\min}}2\right)d\delta_{\min}+\frac12\left[\cot\!\left(\frac{A+\delta_{\min}}2\right)-\cot\!\left(\frac A2\right)\right]dA.
$$
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 $+1.92\times10^{-4}$.
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