Atmospheric Optics & Ice Crystal Refraction: How Hexagonal Prism Geometry and Minimum Deviation Angles Forge Halos, Sun Dogs, and Arcs

1. Outdoor Observer Field Notes: The Luminous Veil of an Approaching Warm Front

To the trained outdoor observer, an impending atmospheric transition rarely announces itself with immediate gale-force winds or sudden downpours. Instead, the earliest harbinger of a mid-latitude cyclonic storm manifests as a subtle, ethereal transformation of the upper sky.

Imagine standing in an open field in mid-autumn. The early morning begins crisp, calm, and crystalline, with surface winds light and variable out of the east-southeast. The barometer rests at a stable 1018 hPa. Gradually, from the western horizon, the deep cobalt blue of the sky begins to soften. It blanches into a milky, translucent sheen that imperceptibly ascends toward the zenith. This is not a chaotic, broken cloud field, but a vast, continuous sheet of Cirrostratus nebulosus, suspended nearly ten kilometres above the earth’s surface.

As the sun climbs through this high-altitude veil, a remarkable optical event occurs. The solar disk, though dimmed slightly to a pearlescent orb, remains sharply defined. Then, precisely at an angular distance of twenty-two degrees from the sun, a delicate ring of spectral light blooms across the sky: the 22-degree halo.

The ring’s inner boundary facing the sun is sharply demarcated and tinged with a faint brick-red hue. Moving radially outward, the colour blends through a pale, yellowish amber into a diffuse, milky-white glare that fades gradually into the surrounding cirrostratus. The patch of sky enclosed within the halo appears distinctly darker than the sky outside it—a striking celestial contrast reminiscent of Alexander’s dark band between primary and secondary raindrops.

       [ SOLAR DISK ]
             |
             |<------- 21.84° (Dark Void: Refraction Forbidden) ------->|
             |                                                          |
             |..........................................................[ RED INNER RIM ]
                                                                        [ VIOLET/WHITE OUTFLOW ]
                                                                        [ EXTENDED SCATTERING ]

Simultaneously, careful field measurements reveal micro-meteorological shifts: - The surface barometer begins an unbroken, steady downward trend (falling at roughly 1.0 to 1.5 hPa every three hours). - High-altitude contrails cease evaporating, stretching from horizon to horizon and expanding into fibratus plumes. - Surface winds gradually back toward the east, while observing the drift of the cirrostratus reveals strong upper-level winds roaring from the west-southwest at 45 to 60 knots.

This luminous geometry in the ice clouds is not merely a visual spectacle; it is a live physical readout of an approaching synoptic warm front, carrying an advective conveyor belt of moisture that will alter the local weather within 12 to 24 hours. To understand why this ring appears with such geometric precision requires journeying into the microphysics of ice crystal habit and the optical laws governing minimum deviation.

2. Microphysics of the Upper Troposphere: Crystal Habits and Aerodynamic Sedimentation

High-altitude cirriform clouds reside within the upper troposphere, typically between 8,000 and 12,000 metres above sea level, where ambient temperatures range from $-30^\circ\text{C}$ down to $-60^\circ\text{C}$. In this desiccated, sub-freezing realm, cloud moisture exists almost exclusively in the solid phase as microscopic ice crystals ($I_h$ hexagonal crystal symmetry).

2.1 Crystal Habits and the Nakaya-Marshall Hierarchy

The precise morphological architecture—or habit—of an ice crystal is governed primarily by two environmental parameters: ambient temperature ($T$) and supersaturation with respect to ice ($S_i$). According to the classical habit diagram established by Ukichiro Nakaya and expanded by modern cloud physics laboratories at the World Meteorological Organization (WMO), ice growth along its crystallographic axes proceeds along two principal vectors: 1. The c-axis: The principal optical and geometric symmetry axis perpendicular to the hexagonal basal planes ${0001}$. 2. The three a-axes: Symmetrically spaced at $120^\circ$ to each other within the basal plane, terminating at the six rectangular prism faces ${10\bar{1}0}$.

When supersaturation is moderate and temperatures hover between $-30^\circ\text{C}$ and $-50^\circ\text{C}$, vapor deposition favors the formation of simple, pristine hexagonal prisms: - Hexagonal Columns: Crystals elongated along the $c$-axis ($c/a > 1$), resembling miniature hexagonal pencils. - Hexagonal Plates: Crystals whose growth along the $c$-axis is suppressed while lateral growth along the $a$-axes dominates ($c/a \ll 1$), yielding thin hexagonal wafers with flat top and bottom basal pinacoids. - Bullet Rosettes & Hollow Columns: Complex polycrystals formed under rapid depositional surges, common in active convective cirrus anvils.

For brilliant, coherent optical halos to form, nature requires pristine, smooth, and optically transparent crystals free of riming, inclusions, or severe structural aggregate defects.

2.2 Aerodynamic Sedimentation and Spatial Orientation

As ice crystals fall through the thin upper-tropospheric air, their orientation is dictated by the fluid dynamics of their descent, described by the crystal Reynolds number ($Re$):

$$Re = \frac{v_t \cdot d}{\nu}$$

where $v_t$ is the terminal fall velocity ($0.1\text{ to }1.0\text{ m/s}$), $d$ is the characteristic diameter ($20\text{ to }500\ \mu\text{m}$), and $\nu$ is the kinematic viscosity of air at high altitudes ($\approx 3.5 \times 10^{-5}\text{ m}^2/\text{s}$).

When small columns and plates tumble in three-dimensional isotropic randomness, sunlight traverses prism faces oriented in every conceivable direction in space. This uniform spatial distribution generates circular, symmetric halos around the light source.

3. Mathematical Foundations: Deriving the 22-Degree Halo via Snell's Law and Minimum Deviation

The formation of the classical circular 22-degree halo is a textbook problem in geometric optics, governed by the interaction of collimated solar rays with the $60^\circ$ prism angle formed by alternate lateral faces of hexagonal ice crystals.

                         RAY TRACING THROUGH A 60° ICE PRISM
                                 Apex Angle A = 60°
                                       /\
                                      /  \
                                     /    \
        Incident Ray                /      \
      ---------------------------->/\      / \
               \ i_1              /  \ r_1/   \
       Normal --\----------------/----\--/-----\-- Normal
                 \              /      \/       \
                  \            /       /\ r_2    \  i_2
                   \          /       /  \        \------------> Emergent Ray
                    \        /       /    \       /
                     \      /       /      \     / Deviation D
                      \    /       /        \   /
                       \  /       /          \ v
                        \/_______/____________\

3.1 Step-by-Step Derivation of Ray Deviation

Consider a light ray traversing a transparent hexagonal prism. The ray enters through one lateral rectangular prism face ${10\bar{1}0}$ and exits through the next alternate prism face. The geometric angle between these two refracting faces is:

$$A = 60^\circ$$

Let: - $i_1$ be the angle of incidence at the first face relative to the surface normal. - $r_1$ be the angle of refraction inside the ice at the first interface. - $r_2$ be the angle of incidence at the interior of the second face relative to its normal. - $i_2$ be the angle of emergence (refraction into air) at the second interface. - $n_{\text{ice}}$ be the refractive index of solid water ice at visible wavelengths ($n \approx 1.309$ at yellow-green $\lambda \approx 589\text{ nm}$). - $n_{\text{air}} \approx 1.000$.

By Snell's Law of Refraction at both interfaces:

$$\sin(i_1) = n \sin(r_1)$$

$$\sin(i_2) = n \sin(r_2)$$

From Euclidean geometry, the internal triangle formed by the prism apex and the refracted ray path dictates that the apex angle $A$ equals the sum of internal refraction angles:

$$r_1 + r_2 = A = 60^\circ$$

The total angular deflection or deviation angle $D$ experienced by the ray is the sum of the deviations at each interface:

$$D = (i_1 - r_1) + (i_2 - r_2) = i_1 + i_2 - (r_1 + r_2) = i_1 + i_2 - A$$

3.2 Finding the Stationary Point of Minimum Deviation ($D_{\min}$)

To determine how light concentrates, we evaluate the first derivative of deviation $D$ with respect to the initial incidence angle $i_1$:

$$\frac{dD}{di_1} = 1 + \frac{di_2}{di_1} = 0 \implies \frac{di_2}{di_1} = -1$$

Differentiating Snell's law at both interfaces yields:

$$\cos(i_1)\, di_1 = n \cos(r_1)\, dr_1$$

$$\cos(i_2)\, di_2 = n \cos(r_2)\, dr_2$$

Because $r_1 + r_2 = A$ is constant, differentiating gives $dr_1 + dr_2 = 0 \implies dr_2 = -dr_1$.

Dividing the two differential expressions:

$$\frac{\cos(i_2)\, di_2}{\cos(i_1)\, di_1} = \frac{n \cos(r_2)\, dr_2}{n \cos(r_1)\, dr_1} = -\frac{\cos(r_2)}{\cos(r_1)}$$

Substituting the condition $\frac{di_2}{di_1} = -1$:

$$\frac{\cos(i_2)}{\cos(i_1)} = \frac{\cos(r_2)}{\cos(r_1)}$$

Squaring both sides and using trigonometric identities $\cos^2(\theta) = 1 - \sin^2(\theta)$:

$$\frac{1 - \sin^2(i_2)}{1 - \sin^2(i_1)} = \frac{1 - \sin^2(r_2)}{1 - \sin^2(r_1)}$$

Substituting $\sin^2(i) = n^2 \sin^2(r)$:

$$\frac{1 - n^2 \sin^2(r_2)}{1 - n^2 \sin^2(r_1)} = \frac{1 - \sin^2(r_2)}{1 - \sin^2(r_1)}$$

Cross-multiplying and simplifying:

$$\left(1 - n^2\right) \sin^2(r_1) = \left(1 - n^2\right) \sin^2(r_2)$$

Since $n \neq 1$, it follows directly that:

$$r_1 = r_2$$

Thus, at the point of minimum deviation, the ray path passes symmetrically through the prism.

3.3 Quantitative Calculation of the 22° Angle

Under this symmetric condition ($r_1 = r_2$ and $i_1 = i_2$):

$$r_1 = r_2 = \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ$$

$$i_1 = i_2 = \frac{A + D_{\min}}{2} = \frac{60^\circ + D_{\min}}{2}$$

Applying Snell’s Law:

$$\sin\left(\frac{A + D_{\min}}{2}\right) = n \sin\left(\frac{A}{2}\right)$$

$$\sin\left(\frac{60^\circ + D_{\min}}{2}\right) = 1.309 \cdot \sin(30^\circ) = 1.309 \cdot 0.5000 = 0.6545$$

Taking the inverse sine:

$$\frac{60^\circ + D_{\min}}{2} = \arcsin(0.6545) \approx 40.8806^\circ$$

Solving for $D_{\min}$:

$$60^\circ + D_{\min} = 81.7612^\circ$$

$$D_{\min} = 81.7612^\circ - 60^\circ = 21.7612^\circ \approx 21.8^\circ$$

(Note: For refractive index $n = 1.310$, common for mid-visible spectrum, $D_{\min} = 21.84^\circ$.)

Ray Deviation Angle D versus Angle of Incidence i_1
  Deviation D (degrees)
    ^
 40 |      \                                 /
 35 |       \                               /
 30 |        \                             /
 25 |         \                           /
21.8|__________\________*________________/____ <- MINIMUM DEVIATION D_min ≈ 21.84°
 20 |                   | (Ray density singularity / Caustic peak)
  0 +-------------------|------------------------->
    0°         20°     40.9°           70°    90°  Incidence Angle i_1

3.4 Why the Inner Rim Is Sharp and the Inside Dark

The differential relationship $\frac{dD}{di_1} = 0$ reveals the physical origin of the halo's appearance: 1. Forbidden Region ($D < D_{\min}$): No geometric ray can pass through a $60^\circ$ ice prism with a total deviation angle less than $21.84^\circ$. Consequently, the sky immediately inside the halo boundary receives no refracted light, creating a distinctly darker disk around the sun. 2. Caustic Intensity Peak: Near $i_1 \approx 40.9^\circ$, the rate of change of deviation is zero ($\frac{dD}{di} \to 0$). Light rays incident over a wide range of entry angles ($35^\circ \le i_1 \le 47^\circ$) emerge bunched tightly together within a fraction of a degree around $21.84^\circ$. This creates an optical caustic—an intense line of brightness forming the sharp inner edge of the halo. 3. Chromatic Dispersion: Ice is dispersive. The refractive index varies across the visible spectrum: - Red ($\lambda = 656\text{ nm}$): $n_{\text{red}} \approx 1.306 \implies D_{\min} \approx 21.54^\circ$ - Yellow/Green ($\lambda = 589\text{ nm}$): $n_{\text{yellow}} \approx 1.309 \implies D_{\min} \approx 21.84^\circ$ - Violet ($\lambda = 404\text{ nm}$): $n_{\text{violet}} \approx 1.317 \implies D_{\min} \approx 22.37^\circ$

Because red light refracts the least, its minimum deviation boundary appears closest to the sun ($21.5^\circ$), giving the inner edge its distinct crimson tint. Blue and violet light refract more strongly ($> 22.3^\circ$), but overlap with the continuous white light emerging from non-minimum deviation angles of other wavelengths, washing out the outer edge into a milky, desaturated white.

4. The Optical Menagerie: Parhelia, 46-Degree Halos, and the Circumzenithal Arc

While tumbling hexagonal crystals produce the uniform 22-degree ring, aerodynamically stabilized crystals acting as oriented prisms project specialized optical phenomena documented in the AMS Glossary of Meteorology and Atmospheric Optics by Les Cowley.

                      THE HIGH-ALTITUDE OPTICAL SKY
                                Zenith
                                  ^
                                  |
                           +--[ CZA ]--+     <- Circumzenithal Arc ("Smile in the Sky")
                          /             \
                         /               \
                        /                 \
                       |     [ 46° Halo ]  |
                       |    +-----------+  |
                       |   /  [22° Halo] \ |
                       |  |   +-------+   ||
                       |  |  /         \  ||
              [Parhelion] +-[*]  [SUN]  [*]-+ [Parhelion / Sun Dog]
                       |  |  \         /  ||
                       |  |   +-------+   ||
                       |   \             / |
                       |    +-----------+  |
                       \                  /
                        \                /
                         +--------------+

4.1 Parhelia (Sun Dogs / Mock Suns)

When flat hexagonal plates have diameters exceeding $30\ \mu\text{m}$, aerodynamic drag forces their basal faces ${0001}$ into a strictly horizontal plane as they drift downward ($Re > 1$).

Top-Down View of Horizontally Oriented Plate:
               Basal Face Parallel to Ground
                      +---------------+
                     /                 \
Incident Ray ======>/   60° Ice Prism   \======> Emergent Ray to Observer
                   +                     +
                    \                   /
                     \                 /
                      +---------------+

Light enters a vertical prism side face ${10\bar{1}0}$ and exits through an alternate vertical prism face. Because the crystal cannot tumble in 3D space, light is refracted exclusively in horizontal planes relative to the crystal: - Low Solar Elevation ($h_s \approx 0^\circ$): The ray path is perpendicular to the vertical face. The minimum deviation is identical to the 22° halo, placing brilliant luminous spots (parhelia) directly upon the 22° ring on either side of the sun. - Elevated Solar Elevation ($h_s > 0^\circ$): As the sun rises, rays traverse the horizontal crystal at an oblique slant. The effective prism apex angle and effective refractive index $n'$ increase according to Bravais' Law of Oblique Refraction:

$$n' = \sqrt{\frac{n^2 - \sin^2(h_s)}{\cos^2(h_s)}}$$

Consequently, as the sun ascends, the parhelia drift outward away from the 22° ring: - At $h_s = 10^\circ$: Parhelia sit at $22.3^\circ$ from the sun. - At $h_s = 30^\circ$: Parhelia shift outward to $24.8^\circ$. - At $h_s = 50^\circ$: Parhelia push past $30.0^\circ$. - Above $h_s \approx 60.7^\circ$: Total internal reflection prevents light from exiting the crystal lateral face, causing parhelia to vanish entirely.

4.2 The 46-Degree Halo: 90-Degree Basal-to-Prism Transitions

Hexagonal columns and plates also possess $90^\circ$ prism corners where a flat basal face ${0001}$ meets a lateral prism face ${10\bar{1}0}$.

Setting the apex angle $A = 90^\circ$ in the minimum deviation equation:

$$\sin\left(\frac{90^\circ + D_{\min}}{2}\right) = n \sin(45^\circ) = 1.309 \cdot \frac{\sqrt{2}}{2} \approx 1.309 \cdot 0.7071 = 0.9256$$

$$\frac{90^\circ + D_{\min}}{2} = \arcsin(0.9256) \approx 67.756^\circ$$

$$D_{\min} = 2 \cdot (67.756^\circ) - 90^\circ = 135.51^\circ - 90^\circ = 45.51^\circ \approx 45.7^\circ \text{ to } 46^\circ$$

Why the 46° Halo Is Rare and Faint

The critical angle for total internal reflection inside ice is:

$$\theta_c = \arcsin\left(\frac{1}{n}\right) = \arcsin\left(\frac{1}{1.309}\right) \approx 49.8^\circ$$

For an apex angle of $A = 90^\circ$, rays have a very narrow window of permissible incidence angles ($i_1 \approx 60^\circ\text{ to }90^\circ$) before encountering total internal reflection at the exit face. Most incident light is reflected internally rather than transmitted, making the 46-degree halo roughly 10 to 20 times fainter than the 22-degree ring, requiring exceptionally uniform and pristine ice clouds to be visible.

4.3 The Circumzenithal Arc (CZA): "The Smile in the Sky"

Often heralded as the most spectrally pure optical phenomenon in the atmosphere, the Circumzenithal Arc appears as an upside-down rainbow centred on the zenith.

                  CIRCUMZENITHAL ARC MECHANISM
                        Sunlight Ray
                             |
                             v
               +-----------------------------+ <- Horizontal Top Basal Face (0001)
               |      Hexagonal Plate        |
               |      Ice Crystal            |
               +-----------------------+-----+
                                       |
                                       +----> Emergent Pure Spectral Ray
                                              (Exits Vertical Prism Face 1010)

• Mechanism: Sunlight enters the horizontal top basal face ${0001}$ of an aerodynamically oriented plate crystal and exits through a vertical prism side face ${10\bar{1}0}$, traversing an exact $90^\circ$ prism angle.
• Solar Height Constraint: Rays can only transmit through this $90^\circ$ geometry without total internal reflection if the solar elevation $h_s$ satisfies:

$$h_s < \arccos\left(\frac{1}{n}\right) + \arcsin\left(\frac{1}{n}\right) - 90^\circ \implies h_s < 32.2^\circ$$

If the sun is higher than $32.2^\circ$, the CZA cannot form. The arc reaches its maximum brilliance and intensity when the sun is at an altitude of $h_s \approx 22.1^\circ$, where refraction occurs at minimum deviation.

5. Synoptic Meteorology in Practice: Forecasting Mid-Latitude Fronts via Optical Tracking

For field researchers, navigators, and outdoor professionals, atmospheric optics provide real-time diagnostic insight into the dynamics of the troposphere. The sequence of optical phenomena reveals the three-dimensional structure of approaching baroclinic waves as described by the classical Norwegian Cyclone Model and modern Warm Conveyor Belt (WCB) theory from the UK Met Office and NOAA National Weather Service.

                      CROSS-SECTION: SYNOPTIC WARM FRONT
Altitude
(km)
 12 +--------------------------------------------------------------------+
    |                                   Cirrus uncinus                   |
 10 |                            Cirrostratus fibratus (22° Halo Forms)  |
    |                     Cirrostratus nebulosus (Halo Peaks)            |  Warm Air
  8 |               Altostratus (Halo Dims / Sun Watery)                 |  Ascent
    |         Nimbostratus (Steady Rain / Snow Begins)                   |  (Warm Conveyor
  6 |       ==========================================                   |   Belt)
    |      /                                                             |
  4 |     / Frontal Inversion Surface                                    |
    |    /                                                               |
  2 |   / Cold Air Wedge                                                 |
    |  /                                                                 |
  0 +--------------------------------------------------------------------+
   0 km            200 km              500 km              800 km       1000 km
   [HEAVY RAIN]    [LIGHT RAIN]        [HALO WASHED OUT]   [BRILLIANT HALO] [CLEAR SKY]
   T = 0 Hours     T + 4 Hours         T + 8 Hours         T + 16 Hours     T + 24 Hours

5.1 The Cloud Optical Depth ($\tau$) Progression

As warm, moist tropical air glides upward along the sloping surface of a retreating cold air dome (a frontal slope typically between $1:100$ and $1:200$), isentropic upglide condenses moisture into a progressively thickening cloud deck:

[Cirrus fibratus]         [Cirrostratus nebulosus]       [Altostratus translucidus]    [Nimbostratus]
τ < 0.1                   0.3 < τ < 1.5                  2.0 < τ < 10.0                τ > 20.0
No Halos                  Optimum Crystal Refraction     Multiple Scattering Dominates Total Extinction
Bare Inversion            Sharp 22° Halo / Parhelia      Halo Washes Out; "Watery Sun" Continuous Precip

1. Optical Initiation ($\tau \approx 0.1\text{ to }0.3$): High Cirrus uncinus ("mare's tails") advance. Contrails persist. Halos are absent due to low crystal concentration.
2. Optimum Optical Coherence ($\tau \approx 0.5\text{ to }1.5$): The sky fills with Cirrostratus nebulosus. Single-scattering dominates. Pristine column and plate crystals produce brilliant 22° halos, sundogs, and circumzenithal arcs. The frontal boundary is approximately 600 to 900 km distant.
3. Multiple Scattering & Extinction ($\tau > 2.0$): The cloud lowers into Altostratus. As optical depth exceeds $\tau = 2.0$, photons undergo multiple random scattering events before reaching the observer. The geometric caustic washes out; the halo blurs into a featureless, diffuse glare.
4. Precipitation Regime ($\tau > 20$): Nimbostratus arrives with continuous stratiform rain or snow. The surface warm front is 50 to 150 km away.

5.2 Calculating Frontal Velocity and Precipitation Arrival

By combining optical observations with local barometric and wind measurements, observers can calculate the arrival time ($t_{\text{precip}}$) of synoptic precipitation:

$$t_{\text{precip}} \approx \frac{\Delta x_{\text{front}}}{v_{\text{advection}}}$$

• Typical mid-latitude warm front propagation speed: $v_{\text{advection}} \approx 30\text{ to }50\text{ km/h}$.
• When a brilliant 22° halo first peaks in a broad cirrostratus sheet, the leading edge of the precipitation-producing altostratus/nimbostratus shield is typically 500 to 700 km upstream.
• Estimated time to precipitation: $12\text{ to }18\text{ hours}$ from the initial peak of the halo, and $6\text{ to }8\text{ hours}$ from the moment the halo degrades into a watery, washed-out sun.

6. Takeaway Box: Today's Meteorological Rule of Thumb

🌤️ Field Rule of Thumb: The Halo-to-Rain Progression

``` OPTICAL SIGNAL BAROMETER WIND BEHAVIOR FORECAST TIMELINE =================================================================================== 1. Sharp 22° Halo Steady (1016 hPa) Light East Rain in 16–24 hrs + Parhelia Cirrus from WSW (65–75% probability)

2. Blurring Halo / Falling 1 hPa/3hr Backing to SE Rain in 8–12 hrs "Watery Sun" Freshening Overcast lowering

3. Corona Appears Falling >2 hPa/3hr Gusty SE/S Precipitation in 2–4 hrs (Water droplets) Rising humidity Secure gear / Shelter ```

Key Distinctions for Field Observers: - Halo vs. Corona: A halo is large ($22^\circ$, span of an outstretched hand at arm's length) with a red inner rim, caused by refraction through ice crystals. A corona is small ($1^\circ\text{ to }5^\circ$, closely hugging the sun or moon) with a bluish inner rim and reddish outer rim, caused by diffraction around liquid water droplets. - The Warning Threshold: If a crisp 22° ice halo transforms into a tight water droplet corona within four hours, cloud base lowering is accelerating rapidly—expect precipitation within six hours.

Authoritative Meteorological & Optical References

1. World Meteorological Organization (WMO) — International Cloud Atlas — The definitive global authority on cloud classification, ice crystal habits, and atmospheric photometeors.
2. UK Met Office — Halos and Atmospheric Optics Guide — Comprehensive overview of frontal dynamics, warm conveyor belts, and optical warning signals.
3. NOAA National Weather Service — JetStream: Atmospheric Optics — Educational physical foundations for optical phenomena and synoptic forecasting.
4. American Meteorological Society (AMS) — Glossary of Meteorology — Technical definitions and peer-reviewed physical equations for atmospheric refraction and cloud physics.
5. Atmospheric Optics by Les Cowley — The reference resource for ray-tracing simulations, crystal aerodynamics, and ice halo mechanics.
6. Wikipedia — 22° Halo Physical Properties — Mathematical formulations and optical characteristics of hexagonal prism refractions.