Dear Kenneth

Thanks for your advise. I found some answers from

https://en.wikipedia.org/wiki/Crystallography

"

• Coordinates in square brackets such as [100] denote a direction vector (in real space).
• Coordinates in angle brackets or chevrons such as denote a family of directions which are related by symmetry operations. In the cubic crystal system for example, would mean [100], [010], [001] or the negative of any of those directions.
• Miller indices in parentheses such as (100) denote a plane of the crystal structure, and regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the (hkl) plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel to [hkl].
• Indices in curly brackets or braces such as {100} denote a family of planes and their normals which are equivalent in cubic materials due to symmetry operations, much the way angle brackets denote a family of directions. In non-cubic materials, is not necessarily perpendicular to {hkl}."

Hope this answer to be right.

FYI:

https://www.revolvy.com/main/index.php?s=Cleavage%20(crystal)

Cleavage, in mineralogy, is the tendency of crystalline materials to split along definite crystallographicstructural planes. These planes of relative weakness are a result of the regular locations of atoms and ions in the crystal, which create smooth repeating surfaces that are visible both in the microscope and to the naked eye.[1]

Types of cleavage

Cleavage forms parallel to crystallographic planes:[1]

• Basal or pinacoidal cleavage occurs when there is only one cleavage plane. Graphite has basal cleavage. Mica (like muscovite or biotite) also has basal cleavage; this is why mica can be peeled into thin sheets.
• Cubic cleavage occurs on when there are three cleavage planes intersecting at 90 degrees. Halite (or salt) has cubic cleavage, and therefore, when halite crystals are broken, it will form more cubes.
• Octahedral cleavage occurs when there are four cleavage planes in a crystal. Fluoriteexhibits perfect octahedral cleavage. Octahedral cleavage is common for semiconductors. Diamond also has octahedral cleavage.
• Rhombohedral cleavage occurs when there are three cleavage planes intersecting at angles that are not 90 degrees. Calcite had rhombohedral cleavage.
• Prismatic cleavage occurs when there are two cleavage planes in a crystal. Spodumene exhibits prismatic cleavage.
• Dodecahedral cleavage occurs when there are six cleavage planes in a crystal. Sphalerite has dodecahedral cleavage

Dear Zhaoyang,

these numbers refer to Miller index notation, as you may have researched. It represents the position of a plan of atoms considering the three spacial axes (X, Y and Z) and each of these numbers is the inverse of the point where the plan of atoms "touches" each spatial axis. The position of the atom plans does not interfere in this notation. So, in the case of 001 plan of clay minerals, it does not matter whether the plan of atoms is positioned on an external crystal surface or in the interlayer basal position.

I hope it helps you.