Cycles, Rings, Strong rings

In chemical graph theory or chemical-related topology a distinction is made between a cycle, a ring, and a strong ring. It is relatively easy to define a cycle:

A cycle is a continuous path (along edges) in which the start and end vertex are the same.

So this means a cycle (also sometimes called a circuit) can be of any size, regardless if there are shorter (or also longer) paths to get back to the ‘home vertex’. We can also say that a cycle is a closed path.

What might be a little more difficult to understand is the formal definition of a ring:

A ring is a cycle that is not the sum of any two smaller cycles.

It is important to emphasize both attributes: two and smaller. The first conclusion is: Okey, in a graph or net there might be rings that are the sum of three, four, five (smaller) cycles, but a cycle is not a ring if it is the sum of exactly two (smaller) cycles.

In the following figure the graph of a cube is shown and some cycles (highlighted in red) within this graph:

Which of these cycles are rings? I think it is clear, that the two cycles in the top row are rings. Furthermore, the cycle in the middle in the bottom row is a ring, while the left and right cycle are not rings.  Why? Well, the 6-cycle on the left is the sum of two smaller 4-cycles. The 6-cycle in the middle is the sum of three(!) smaller cycles but not the sum of two smaller cycles. And the 8-cycle on the right can be considered as the sum of a 4-cycle and a 6-cycle (the latter can be formed as the sum of two smaller 4-cycles, the top-most and the central one):

Well, but shouldn’t it be possible to build the 6-cycle in the middle as the sum of a 4-cycle (top-most) and a 6-cycle (which itself be the sum of the central and right 4-cycles)? Yes, that is correct, but the definition states that the cycles of a sum of cycles have to be smaller! This is not the case here!

Interestingly, there is an alternative definition of a ring which is, at least for me, much more intuitive and comprehensible:

A ring is a cycle that contains along its path no potential short-cut (of the length of exactly one edge) to the ‘home vertex’.

It is easy to see that the cycle on the left has one short-cut and the cycle on the right even two (dashed edges):

So, what are now strong rings? They can be defined as follows:

Strong rings are rings that are not the sum of any number of smaller cycles.

From that definition It follows that only the two cycles in the top row are strong rings.

I would like to thank Michael Fischer (@ZeoliteMiFi) for helpful discussions.

 

 

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Strengite

Strengite

  • Named after Johann August Streng (1830-1897), German mineralogist, University of Giessen, Germany
  • Formula: FePO4 · 2 H2O
  • Space group: Pbca (No. 61)
  • Crystal system: orthorhombic
  • Crystal class: mmm
  • Lattice parameters: a = 8.722 Å, = 9.878 Å, c = 10.8117 Å, αβ = γ = 90°

Picture: Christian Rewitzer  | CC BY-SA-3.0


Crystal structure (click on the picture to download the VESTA file):

(K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,”J. Appl. Crystallogr., 44, 1272-1276 (2011).)

  • FeO6 octahedra (dark-red)
  • POtetrahedra (green-blue)
  • Oxygen (red)
  • Hydrogen (white)

For a 3D interactive version on sketchfab, see here:

https://skfb.ly/6BVJs

Spessartine

Spessartine

  • Named after its type locality Spessart (Bavaria, Germany)
  • Formula: Mn3Al2[SiO4]3
  • Space group: Ia-3(No. 230)
  • Crystal system: cubic
  • Crystal class: m-3m
  • Lattice parameters: a = = c = 11.621 Å, αβ = γ = 90°

Picture: Rob Lavinsky – iRocks.com  | CC BY-SA-3.0


Crystal structure (click on the picture to download the VESTA file):

(K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,”J. Appl. Crystallogr., 44, 1272-1276 (2011).)

  • MnO8 polyhedra (purple)
  • AlO6 octahedra (blue)
  • SiOtetrahedra (orange)

For a 3D interactive version on sketchfab, see here:

https://skfb.ly/6BQQP

Celestial Celestine

Celestine

  • named from the Latin word caelestis meaning celestial, which in turn is derived from the Latin word caelum meaning sky or heaven because of its often soft blue color
  • pure Celestine is colourless
  • due to lattice defects in Celestine, colour centres are created which give the crystal its characteristic bluish colour
  • these centers are often additionally stabilized by the presence of pottasium ions
  • heating to over 200 °C “cures” these lattice defects and the mineral loses its color
  • radiation with X-rays creates new or more lattice defects and the color returns or can be intensified.
  • Formula: SrSO4
  • Space group: Pnma (No. 62)
  • Crystal system: orthorhombic
  • Crystal class: mmm
  • Lattice parameters:  a = 8.360 Å, b = 5.352 Å, c = 6.858 Å, αβγ = 90°

Picture: Rob Lavinsky, iRocks.com – CC-BY-SA-3.0


Crystal structure (click on the pictures to download the VESTA file):

(K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,” J. Appl. Crystallogr., 44, 1272-1276 (2011).)

  • SO4 tetrahedra (yellow)
  • SrO8 polyhedra (green)
  • Oxygen (red)

For a 3D interactive version, see here:

https://skfb.ly/6zG6R

Porous materials, minerals and bees

Thoughtful and very well written. Nice analogy with the bees!

Molecular Dreams

Overwhelmed with the increasing flow of new scientific discoveries and related literature? You’re not alone. We live in the information overload era: too much to read, too little time, and life is short. Probably we’d need more readable, shorter papers too. Why writing a long one? Perhaps, it might connect disciplines which speak different languages but have much in common. Like material science and mineral science.

Let’s start from the first one.

You can make materials for solar cells, optical devices or medical sensors by trapping molecules or nanoparticles inside a “host”. Once there, molecules are no longer free to move, like in a gas or a liquid.  This process, called “confinement”, brings to life new properties, which were not present in the individual molecules and are very useful in technology.  Energy transfer or information storage, for instance, are made possible by the organization of the confined molecules

P1270522 The…

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Ice II (Ice-two)

Ice II (Ice two)

  • can be formed from hexagonal Ice (Ice Ih) at 198 K and 3000 bar or by decompressing Ice-five (Ice V) at 238 K
  • Ice II is likely to be a major rock-forming mineral in the outer Solar System
  • It may form a major proportion of icy moons such as Jupiter’s Ganymede
  • Density: 1.16 g/cm3

Structural features:

  • Ice-two is a proton-ordered form of ice
  • there are two types of 6-membered rings; one is almost flat (Type A) the other one has a more puckered, chair-like conformation (Type B)
  • these two types of rings are strictly alternating stacked along the c axis

  • If you look along the c axis, you will see that the two types of 6-rings are slightly rotated against each other (~ 16 degrees)

  • Space group R-3
  • Lattice parameters:
    • a = b = 12.935 Å, c = 6.233 Å
    • α = β = 90°γ = 120°

  Here, you can download the CIF.

[Atomic structure figures created with VESTA:
K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,” J. Appl. Crystallogr.44, 1272-1276 (2011).]

Ice Ic

Ice Ic ( Ice one cubic)

  • can be formed by condensation of water vapour at reduced pressure well below -80 °C
  • is metastable with respect to Ice Ih (approx. + 50 J/mol)
  • it seems to be very difficult to grow large phase-pure Ic ice crystals; they contain, to a certain extent stacking-disordered or hexagonal ice

Structural features:

  • like hexagonal ice cubic ice is a proton-disordered phase
  • every water molecule is involved in 4 H-bonds (2 acceptors, 2 donors)
  • tetrahedrally coordinated
  • O-D bond length approx. 101 pm
  • D …. O-D distance approx. 174 pm
  • O … O distance approx. 275 pm
  • 6-membered rings (exclusively chair conformation)

  • sometimes also called “cristobalite ice” because the oxygen atoms occupy the Si analogeous positions in the SiO2 phase cristobalite
  • it would be also justified to call it diamond-like ice 🙂
  • Space group Fd-3m
    • a = b = c = 6.3510 Å
    • α = β = γ = 90°

 

  Here, you can download the CIF.

[Atomic structure figures created with VESTA:
K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,” J. Appl. Crystallogr.44, 1272-1276 (2011).]