= Cherenkov cone angle
{c}
{title2=$\cos\theta=1/[\beta n(\omega)]$}
Let a <charged particle> travel along $+z$ at constant <speed> $v=\beta c$, with $0<\beta<1$. For a <plane electromagnetic wave> with <wave phase> $\mathbf k\cdot\mathbf r-\omega t$, two points along the track separated by time $\Delta t$ contribute with phase difference
$$
\Delta\phi=(kv\cos\theta-\omega)\Delta t,
\qquad k=\frac{n(\omega)\omega}{c}.
$$
Here $\theta$ is the <angle> between the wave's <wave vector> and the forward particle track. Constructive addition along a long uniform track requires $kv\cos\theta=\omega$. Thus
$$
\cos\theta=\frac{1}{\beta n(\omega)},\qquad
\beta n(\omega)>1.
$$
A real forward cone with nonzero opening exists only above this threshold. At $\beta n=1$ the cone closes and the ideal <Frank-Tamm formula> gives zero yield. For $\beta n<1$ there is no allowed <angle>, rather than an imaginary observable cone. The requirement is <phase matching>; substituting the <group refractive index> would describe a different condition.
There is a second geometric derivation when $n$ is constant. In time $\Delta t$, the charge advances $OP=v\Delta t$, while an earlier spherical phase front has radius $OQ=c\Delta t/n$. Its tangent envelope passes through the current position $P$. The radius $OQ$ is normal to the envelope, so the right triangle gives $\cos\theta=OQ/OP$. Rotating about the track makes a cone. Its surface half-angle $\eta$ is complementary to the emission <angle> $\theta$, not equal to it.
\Image[/cherenkov-wavefront-cone-geometry.png]
{title=Original <Cherenkov radiation> geometry: phase fronts, their tangent cone, and the emission normal}
{description=Chosen constant <refractive index> $n=1.50$ and particle <speed> $v=0.90c$. The circles are phase fronts at one time from successive earlier track positions. The orange normal makes $\theta\simeq42.21^\circ$ with the track; the tangent cone makes $\eta\simeq47.79^\circ$. This is a nondispersive geometric construction, not a measured particle track or a broadband pulse envelope.}
In a dispersive medium, apply the <phase matching> derivation separately at each <angular frequency>. The circle construction uses one fixed phase speed; it does not determine the envelope of a broadband <wave packet>. In particular, <group velocity> controls packet propagation under its usual narrow-band assumptions, and the simple complementary-angle construction need not describe a dispersive pulse front. A finite track also broadens the ideal directional selection: adding contributions over duration $T$ gives a factor proportional to $\sin(\Delta T/2)/\Delta$, with $\Delta=kv\cos\theta-\omega$, rather than an infinitely sharp direction.
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