= Solution
<Snell's law> is $n_1\sin\theta_1=n_2\sin\theta_2$, with angles measured from the interface normal. The <Huygens-Fresnel principle> explains it by constructing the transmitted <wavefront> from secondary wavelets: during the same time interval, distances travelled in the two media are proportional to $c/n_1$ and $c/n_2$. The geometry of the advancing plane <wavefront> then gives $\sin\theta_1/\sin\theta_2=(c/n_1)/(c/n_2)=n_2/n_1$.
For incidence from index $n$ into vacuum, the transmitted angle is $\alpha=\arcsin(n\sin\theta)$. The ray bends away from the normal, so its positive deviation is
$$
\boxed{\delta(\theta,n)=\arcsin(n\sin\theta)-\theta.}
$$
For $0<\theta<\theta_c=\arcsin(1/n)$,
$$
\frac{\partial\delta}{\partial\theta}=\frac{n\cos\theta}{\sqrt{1-n^2\sin^2\theta}}-1>0.
$$
Thus the graph rises monotonically from zero to a finite endpoint with a vertical tangent:
$$
\boxed{\theta_c=\arcsin(1/n),\qquad \delta_{\max}=\frac\pi2-\arcsin(1/n).}
$$
At the endpoint the transmitted ray grazes the interface. Beyond the <critical angle for total internal reflection>, no propagating transmitted ray exists and <total internal reflection> occurs.
Inside a <prism>, the two ray angles with the face normals sum to its apex angle. For the equilateral cross-section, they are $\theta$ and $\pi/3-\theta$. The entry deviation equals the corresponding exit deviation under ray reversal. Adding the two changes of direction therefore gives
$$
\boxed{\Delta(\theta)=\delta(\theta,n)+\delta(\pi/3-\theta,n).}
$$
This is symmetric about $\theta=\pi/6$. Symmetry alone gives a stationary point; to establish that it is a minimum, differentiate once more:
$$
\frac{\partial^2\delta}{\partial\theta^2}=\frac{n(n^2-1)\sin\theta}{(1-n^2\sin^2\theta)^{3/2}}>0
$$
within the transmitted-ray interval. The sum is therefore strictly convex, and
$$
\boxed{\theta_{\min}=\frac\pi6,\qquad \Delta_{\min}=2\arcsin(n/2)-\frac\pi3.}
$$
These expressions require $1<n<2$ for a non-grazing path through both faces; $n=2$ is a grazing limit, and $n>2$ admits no such two-face transmitted path. More explicitly, both internal angles must be at most $\theta_c$, which is possible only if $2\theta_c\geq\pi/3$.
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-prism-deviation.png]
{title=Single-interface deviation and the symmetric minimum of an equilateral prism, illustrated for index 1.5}
Atmospheric hexagonal ice crystals can present pairs of faces with a $60^\circ$ angle, acting as small <prisms>. Their orientations turn the minimum-deviation cone around the Moon into an <optical halo>. No transmitted ray of this family is deflected less than the <minimum deviation of a prism>. Moreover, near the minimum, $\Delta-\Delta_{\min}$ is quadratic in the departure from the symmetric ray, so a range of incident directions is compressed into a small outgoing angular range. \b[This <caustic> makes the inner rim sharp and bright], with the ray density decreasing farther outside. The common <22-degree halo> has an angular radius near $22^\circ$ about either the Sun or Moon, hence a diameter near $44^\circ$.<International Cloud Atlas by World Meteorological Organization>[Section 3.2.3.1.1: radius, central luminary, bright ring and darker interior]{ref}
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