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How do atoms work and what shape do they have ?

For a quick overview (like given on this page) a detailed understanding of the inner workings of atoms is not (yet) necessary. For more detailed analysis (not given on this page) a detailed understanding is absolutely indispensable. If you want to dive down a bit further right away then check out the main article: "The nature and shape of atoms"

The focus here will be limited on the lightest, simplest, most common, and for APM most relevant elements which are situated at the upper right corner of the periodic table. What will not be discussed here are heavier elements like transition metals and beyond which can have quite complicated electron configurations.

Tetrapodal arrangement of electron clouds -- that form then bonds between atoms

Tetrapodal electron cloud arrangement (tech term: sp³) Note: This is still a simplified view. The orbitals are actually more bean shaped, overlapping and blurry. But depicting it more realistically would obfuscate the geometric arrangement too much.

The outermost electron clouds are the most relevant since usually only these take part in covalent chemical bonds. Bonds that link together molecules or crystals or crystolecules. In the elements of concern on this page (see above) the most common arrangement of the outer "electron clouds" is tetrapodal (aka tetrahedral). There are with four lobes (four==tetra). The geometry is depicted in blue here. Note that the disconnected small lobes and the big lobes that are on exactly 180° opposing sides belong together respectively forming a single electron cloud each.

A more technical term for "electron clouds" is "orbitals". The orbitals that are depicted in blue here are called "hybrid orbitals".
"Hybrid" means that one does not get the shape of the electron clouds directly as solutions out of the underlying math (the wave equation) but that instead one needs to add up some of the electron cloud shapes solutions to get the the electron cloud shape that is actually physically present.

• Orbitals filled with two electrons from the host atom (as depicted here) are called "lone pairs" and repel lone pairs from other atoms.
• Orbitals filled with one electron from the host atom and one electron from a neighbouring atoms merge together and are called bonding "molecular orbitals".
• In case too many electrons are missing (sometimes the case with elements to the left of carbon in the periodic table) the geometric arrangement of electron clouds can change. Details elsewhere.

The four orbitals in the tetrapodal geometry do not lie in a common plane (they are not coplanar).

Which elements make good "construction kit building blocks" and why

• Elements with too many electrons in the outermost shell (more than four of the maximum eight) have them forming lone pairs. These can no longer form really strong covalent bonds. This makes it harder to impossible to form three dimensional networks with these elements only. One may get only sheets or chains instead. Repulsive electron cloud lone pairs are useful though everywhere where we do want surfaces where almost nothing shall ahrere to. (See: Passivation). There are some cheavats with weaker coordinative bonds.
• Elements with too few electrons in their outermost shell (less than four) may have their orbitals rearrange themselves into geometries that are different from tetrapodal. This too can make it harder to form three dimensional networks with these elements only. Prime example: The element Boron.
• Elements with exactly the right amount of electrons (exactly four) like e.g. carbon (and silicon) can bond to other atoms in all four non-coplanar tetrapodal directions and can thus form three dimensional crystal structures on their own. Not just sheets or chains. Prime examples for tightly meshed 3D networks of that kind are diamond crystals, silicon crystals and silicon carbide (aka moissanite) crystals.

Carbon:

• Unlike the elements to its right (starting with nitrogen) with more electrons it can do can create strong covalent 3D networks on its own.
• Unlike the heavier elements below it (starting with silicon) it can still form other other bonding geometries too (see below).
  
• Unlike the element to its left (boron) it has enough electrons in its outermost shell to maintain a tetrapodal geometry of its bonds.

There are essentially the reasons why carbon carbon is sometimes referred to as "king of all the elements".

Triangular arrangement of hybrid orbitals

Triangular electron cloud arrangement. (tech term: sp²) Note: the fourth orbital is not depicted!

There are other possible electron cloud arrangements too. The second most common one is triangular (as depicted here in blue).

The light elements have four outer shell orbitals but only three are depicted here, so one is obviously missing. The missing/non-depicted fourth one sticks out vertically both up and down equally from the image-plane. The fourth orbital has no small and big lobe like the three depicted hybrid orbitals have. It is a fundamental orbital (a p-orbital - a raw solution from the underlying math) with two lobes of exactly equal size.

The three hybrid orbitals lie in the same plane and the fourth (the fundamental) orbital has no preference for facing upwards or downwards. Thus atoms that take on this triangular orbital arrangement cannot form three dimensional structures in a way like the atoms with tetrapodal structure. Instead they can only form two dimensional sheets.

Just like in the tetrapodal arrangement also in the triangular arrangement case carbon (and silicon) atoms have the ideal number of electrons to neither form lone pairs (repulsing other lone pairs) nor change orbital arrangement. Thus a prime example for sheets out of atoms in triangular orbital arrangement are sheets made out of carbon.

In the simplest, that is fully planar, form this is called a graphene sheet. Stacks of large graphene sheets form very hard single crystalline graphite. Normal pencil mine graphite is polycrystalline allowing the small sheet-flakes to slide over each other making it very soft. (Side-note: Larger chunks of single crystalline graphite do not occur naturally but can by synthesized today. It is called: HOPG)

In a graphene sheet the fourth orbital (the non-hybridized fundamental p-orbital) different from the three triangularly arranged hybrid orbitals plays a very special role. Not only sticks it out both sides equally it also shares one bond in three directions simultaneously on each side. All those doublesided p-prbitals fuse together to one single giant (double sheeted) molecule orbital spanning over the whole sheet on both sides. This allows electrons to move freely (electric conductivity).

Bending graphite sheets by various means can drastically (and usefully) change the electronic properties. From semi-conductivity to very high conductivity (much better than copper or silver).

The tech term for hybrid orbitals that assume the here describesd triangular shape is: sp².

Beside electronic property changes bending sp² sheets (graphite or other) also allows them to form three dimensional structures when the sheets locally can actually only be two dimensional.

• Flat sheets must have all atoms arranged in hexagons.
• Flat sheets occur rolled up into tubular shapes (Nanotubes in general). Beside various diameters different rolling angles are possible (causing different eletronic properties).
• Sheets can be bent convex (or concave depending on the onlooking side) by replacing some hexagons with pentagons, squares or even triangles (Buckyballs in general).
• Sheets can be bent hyperbolic by replacing some hexagons with heptagons, octagons, ... (Foam like structures e.g. DLC)