2. Fundamental patterns of snow crystals
Many researchers[11] have carried out experiments on the growth of artificial snow crystals since Nakaya. Kobayashi[12] consolidated them and updated the Nakaya diagram of snow crystal shapes as a function of the temperature and the excess vapor density(Fig. 2). The diagram clearly shows that there are two kinds of changes of snow crystal shapes. The first one is that the basic patterns of snow crystals alternate three times with falling temperature: Namely, from plate to prism at －4oC; to plate again at －10oC; and then to prism at － 22oC. This temperature dependent change is the so-called habit change. The second is that the patterns change from the simple hexagonal shape to the more complicated shape with an increasing supersaturation. For example, from a hexagonal plate to a sector plate and then to a hexagonal dendrite in the temperature ranges between 0 and －4oC and between － 10 and －22oC; or from a hexagonal column to a skeletal crystal and then to a needle crystal in the other temperature ranges. This change relates to the morphological instability occurring during the growth of polyhedral crystal [13].
Fig. 3 gives an illustration showing two basic changes of snow crystal patterns. The horizontal axis indicates the temperature dependence of the habit change, and the vertical axis indicates the development of morphological instability as a function of supersaturation. As a result, we claim that the shapes of snow crystals should be distinguished by only four categories, delineating the growth features of snow crystals. Namely they are the plate(a), the prism(b), the dendrite(c) and the needle(d). Natural snow crystals may be formed as intermediate shapes between different types and/or the complete transition from one type to the other type during the growth. Consequently, discussion of the formation mechanisms of various patterns on the snow crystals is summarized by the mechanisms for two fundamental changes shown in Fig. 3.
On the other hand, though it is well known that the bulk phase diagram of ice includes more than 10 different crystallographic structures or polymorphs, it should be noted that the complicated changes observed in snow crystal morphology are never related to these crystallographic phase transitions. Since the temperature and pressure ranges in which the snow crystals grow are extremely narrow and completely included in the region of hexagonal ice I(h), it is sufficient for the following discussion to refer the crystal structure of ice I(h) [14]. Fig. 4a gives a schematic illustration for the arrangement of water molecules. In Fig. 4a, the open spheres indicate the oxygen atoms, and the thin solid lines which are connecting each of the oxygen atoms indicate the hydrogen bonds. The arrangement of oxygen atoms in the ice I(h) is categorized as the wultzite structure (P63mmc). The hydrogen atoms (indicated by solid spheres) are arranged between each oxygen atom in such a manner as to satisfy the ice rules or the Bernal-Fowler rules. Corresponding arrangements and indices of crystallographic axes and faces are shown in Fig. 4b. It goes without saying that this hexagonal arrangement of water molecules relates to the hexagonal symmetry of snow crystals.
Before considering the basic pattern formation mechanisms of snow crystals, it will be very useful to consider the initial process of snow crystal formation in the cloud. Small ice particles (the origin of snow crystals) appear in the cloud by freezing of small cloud droplets with the diameter of about 10μm. Such particles with spherical shapes start to grow in the supersaturated water vapor. Fig. 5 shows a sketch of growth process of an ice particle [15]. Since only the crystallographic planes with the lowest growth velocities can survive during the growth, the initially spherical particles (A in Fig. 5) grow into the hexagonal prisms (H in Fig. 5) with an aspect ratio of nearly unit (lc/la ～1).
The fundamental prisms continue to grow while falling through the cloud, and their shapes change into one of the plate-like or prism-like habits. Here it should be noted that the habit of a snow crystal is determined as a necessary consequence of crystal growth. When the growth rate of the basal {0001} plane is larger than that of the prism {1010} planes (namely, R(0001)＞R(1010)), the habit becomes prism-like (lc/la＞1). When the reversed relation R(0001)＜R(1010) holds, the plate-like habit (lc/la＜1) appears as shown in Fig. 5. Consequently, the fundamental challenge in understanding the habit change is to clarify the alternative changes in the growth rates between the basal and prism planes.