# The monoclinic crystal system and the skew angle beta

One of the things we have tried to teach in our course is:

It is not the metric that determines to which crystal system a crystal belong. It is the other way round – the symmtry of the crystal determines the metric, although not in a biunique way.

In many textbooks the metric for the monoclinic crystal system is given as:

a ≠ b ≠ c, and αγ = 90°, β ≠ 90°  ;

And this is simply wrong. The correct statement is that there are no restrictions concerning a, b, and c, and there is also no restriction regarding the angle β, hence:

αγ = 90° – that’s it.

The important message here is that it is indeed not very likely that beta equals 90°, but it is not (mathematically) forbidden, the angle β could ‘accidentally’ be 90°.

While it is one thing to show mathematically that in the monoclinic crystal symmetry the maximum symmetry is 2/m and that therefore the angle beta can also be 90° it is another question if nature produces indeed a crystal belonging to the monoclinic crystal system and nonetheless possesses an angle β of 90°. This could then be considered as a sort of empirical proof.

Here it is 🙂

# Crystal structure of sodium potassium zinc diphosphate NaKZnP2O7

Yu.F. Shepelev, A.E. Lapshin, M.A. Petrova

Journal of Structural Chemistry

November 2006, Volume 47, Issue 6,

Space group P21/n, a = 12.585(5) Å, b = 7.277(5) Å, c = 7.428(5) Å, β = 90.00(5)° (!)

And this is how it looks like:

PO4 tetrahedra (purple)
ZnO4 tetrahedra (grey)
sodium (yellow)
pottassium (blue)