A re-visited 13° halo observation from 2013, and some thoughts about the responsible crystal faces

Circular halos of 12°-13° in radius are named “exotic” because they do not fit in the (nowadays) traditional sequence of well-documented halo radii from pyramidal ice crystals (9°, 18°, 20°, 22°, 23°, 24°, 35°, 46°). The first known photographs of such a halo were obtained at the South Pole, December 11th-12th, 1998, by Walter Tape, Jarmo Moilanen and Robert Greenler. Up to now, there are only few more (Michael Theusner, Bremerhaven, October 28th, 2012; Nicolas Lefaudeux, Paris, May 04th, 2014).

When Marko Riikonen and Marko Pekkola discussed this exotic halo in their presentation at the Light & Color meeting in Granada (May 31st, 2016), I was reminded of a photographic observation of my own from Dresden, March 26th, 2013. It happened during a rare halo activity outburst which resulted in a pretty busy week full of taking and analyzing photographs for me (the original report can be found here). I even mentioned the strange circle of about 12° in radius in the report, and, privately, played with the thought that it might be the same kind of halo as the one from the 1998 South Pole observation (the Bremerhaven case from 2012 had, unfortunately, slipped my attention). However, it seemed more justified to attribute it in public to an artifact back then. In contrast to this, I am now, with hindsight of 2016, convinced that I actually captured an exotic 13° halo. In addition to the older photos, I assembled a sum stack from 25 individual frames, re-centered on the sun and covering the time interval from 15:22-15:34 CET:

original stack:2013_03_26_1522_1534_stack
unsharp masked, 13° halo marked by arrow:2013_03_26_1522_1534_stack_usm_mark
red color channel minus blue color channel:2013_03_26_1522_1534_rb_gam_usm_mark
For the explanation of this halo, it is necessary to introduce additional crystal faces beyond the traditional basal {0001}, prism {10-10} and pyramidal {10-11} ones. The symbols in brackets are the crystallographic Bravais-Miller indices as described in the “Angle X” book of Tape and Moilanen, chapter 9. Curly brackets indicate sets of symmetrically equivalent faces (e.g. the two basal faces, six prism faces or twelve pyramidal faces, respectively), while round brackets denote one specific face out of such a set. By convention, a negative number is indicated by an overlined symbol, but due to typographical restrictions I will use the common minus sign here. A refracting wedge is formed by the combination of two non-parallel faces and will give rise to a halo through minimal deflection as long as the wedge angle does not exceed a certain threshold (twice the critical angle of total internal reflection).

Nicolas Lefaudeux demonstrated that the numerous exotic halos (mostly plate arcs) from the Lascar display can be simulated with crystal shapes involving basal, pyramidal and {30-32} faces over a wide range of solar elevations. This also included a 13° plate arc (from a (30-32) → (-3032) face combination: wedge angle 39.0°, minimum deflection 12.9° for n = 1.31), thus providing an explanation for the 13° halo if such crystals would exhibit random orientations instead of plate-like ones.

However, one hast to keep in mind that the higher the Bravais-Miller indices, the less likely is the formation of a specific type of faces (a consequence of the so-called “Bravais law” from crystallography). The Lascar display has been unique (as of now), and the large range of plate arcs, whose variable shapes had been documented for a large range of solar elevations, allowed Nicolas to identify the relevant {30-32} face type. It can, nonetheless, not be ruled out that other 13° halos may be caused by more common face combinations with lower indices.

To get an overview of the parameter space, I played around a bit with the Bravais-Miller indices and found that the amount of wedge angles (and thus possible halo radii) increases tremendously when the index numbers are allowed to climb up successively. Using a small Matlab script it was quite easy to identify equivalent combinations (those with identical wedge angles) and to throw out all wedges with angles too large for halo formation. To simplify matters a bit, I will restrict the discussion to the most likely candidates beyond the traditional faces: more inclined pyramidal faces {10-12} (“Angle X” p. 96, Fig. 9.6) and second-order pyramidal faces {11-21} (“Angle X” p. 96, Fig. 9.7).

Using the crystallographic c/a ratio of 1.63 for ice, I calculated the following table:

no. of face combination faces involved wedge angles [°] halo radii [°]
(n = 1.31)
1 {0001}
{10-10}
basal, prism 60.0
90.0
21.8
45.7
2 {0001}
{10-10}
{10-11}
basal, prism, “traditional” pyramids
28.0
52.4
56.0
62.0
63.8
80.2

8.9
18.3
19.9
22.9
23.8
34.9
3 {0001}
{10-10}
{10-11}
{10-12}
basal, prism, “traditional” pyramids, more inclined pyramids
18.8
40.1
43.3
46.7
49.9
70.0
72.8
74.7
86.5
87.8
92.2
93.5

5.9 (a)
13.3 (b)
14.5
15.9
17.2
27.4 (c)
29.3
30.6
41.2
42.7
49.3
51.6
4 {0001}
{10-10}
{10-11}
{10-12}
{11-21}
basal, prism, “traditional” pyramids,
more inclined pyramids,second-order pyramids

29.7
34.1 (2x)
38.7
53.6
57.1
68.2
69.3
72.9
77.7
82.1
97.9

9.5
11.1
12.7 (d)
18.8
20.4
26.3
27.0
29.3
32.8
36.6
64.3

The dots “…” indicate that all wedge angles and halo radii from the upper rows have to be added to the respective list, i.e. face combination 1 (basal and prism) gives the familiar two radii (22° and 46°), face combination 2 (basal, prism, traditional pyramids) gives 8 halos (the familiar pyramidal radii), etc. I did not include the wedge angle of 99.8°, which is slightly above the threshold for n = 1.31.

Most interesting are the ray paths marked (a)-(d), as they reasonably match previously observed exotic halos. In more detail, path (a), e.g. realized by a (10-11) → (-101-2) transition (this is just one out of several symmetrically equivalent options) will create a halo of 5.9° in radius, which also has been photographed during the South Pole display of December 11th-12th, 1998. For the 13° halo, ray paths (b) and (d) are possible candidates ((10-12) → (-110-2) and (10-12) → (-211-1), respectively). Path (c) ((10-10) → (-1102)) will create a halo close to 28° (“Scheiner’s halo”) in radius. There are hints of it in the South Pole photographs (presented here) and it also was observed by Jari Luomanen on ice-crystal covered snow on April 07th, 2012, in Kontiolahti. It also occurred in the Lascar display, but can in this case be readily explained using the {30-32} faces, whose presence had been previously established through the analysis of the exotic plate arcs.

In fact, the “Lascar faces” {30-32} combined with ordinary basal, prism and pyramid faces do provide suitable explanations not only for the 28° and 13° halos, but also for the 6° halo. The purpose of my calculations is not to discard this option, but to illustrate that there are many other face combinations or “refracting wedges” that give similar halo radii, and even with optimal angular calibration it will not be possible to discriminate between them from photographs. Unless there are strong reasons from ice crystal growth physics why only {0001}, {10-10}, {10-11}, and {30-32} faces will be developed on crystals in the atmosphere, we should not forget about Bravais’ law and other, intuitively more likely crystal habits. In consequence, it seems impossible to solve the puzzle of determining the actual crystal shape from the observation of circular exotic halos alone. In my opinion, only from the very rare displays (like Lascar) during which non-random orientations and their resulting arcs can be observed for a broad range of solar elevations, some sound conclusions can be drawn.

4 thoughts on “A re-visited 13° halo observation from 2013, and some thoughts about the responsible crystal faces

  1. You mean the excotic pyramid is on top of the normal pyramid (or the other way around)? Not like the combination pyramids in Nicolas Lascar paper where the bottoms of normal and excotic pyramids are facing each other? Long time ago Walt to Ursa office a long fax with different theoretical pyramidal crystals where the pyramids were on top of each other. I wonder if it still exists somewhere.

    • Yes, the easiest (and likely the only) way for such a crystal with all these faces would be two blunter pyramid caps (with {…2} and {…-2} faces + basal ends) at the top and bottom of an ordinary pyramidal crystal. A drawing would be helpful, but it would take me some time to prepare something suitable.

  2. What’s the apex angle of the exotic pyramid in 3. of your table? This scenario could be easily tried with Jukka’s sotware.

    • Hi Marko,

      “Angle X” mentions that the apex angle should be 46.7° in this case. But “combination 3” is more than just a pyramidal crystal with a different apex angle. There are also all faces from the ordinary 28° pyramids included. (a) and (c) are examples for “inter-pyramid” ray paths.

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