Simple Monoclinic Unit Cell


The monoclinic unit cell has three lattice parameters a, b, and c that each have different lengths; two vector angles α and γ are at 90º to each other, and the third vector angle β is not.

Since simple monoclinic is one of the less-common crystal structures, I’m assuming someone searching for this is a somewhat advanced student in materials science, so I’ll give you the facts quickly and with little explanation. If you are new to materials science and don’t know what something means, you can check out my articles on the FCC, BCC, or HCP crystals, which explain terminology more slowly.

Although there are many crystal structures that fit with “monoclinic” symmetry, the simple monoclinic crystal structure (abbreviated SO in this article) has exactly 1 atom per lattice point in the primitive monoclinic Bravais lattice.

As far as I know, there are no real-world materials that exhibit a simple monoclinic unit cell. If this structure were to exist, it would be like a tilted box with 3 different side lengths, with an atom on each corner. There are “primitive monoclinic” crystals based on the Bravais lattice, which exist with multiple atoms that overall displays primitive monoclinic symmetry. The simple monoclinic unit cell would belong to space group #10 or P2/m, with Pearson symbol mP1. There is no prototype or Strukturbericht for simple monoclinic, since it does not exist with real atoms.

The simple monoclinic Bravais lattice can be imagined as a titled box with 3 different side lengths, with a lattice point on each corner. Pure materials never take this crystal structure, and it exists only mathematically. Simple monoclinic has 1 atom per unit cell, lattice constants a, b, and c, lattice angles α=γ= 90º and β ≠ 90, Coordination Number CN=2, and Atomic Packing Factor APF=\frac{4\pi r^}{3abc\cdot\sin{\beta}}.

Common Examples of Simple Monoclinic Materials

None.

However, there are still materials made of multiple kinds of atoms (or a multi-atom basis) that have a primitive monoclinic Bravais lattice, such as martensitic NiTi, alphaPu, and β-Se.

Simple Monoclinic Coordination Number

You can think of simple monoclinic as a distortion of the simple monoclinic crystal, by tilting along one axis. Simple monoclinic has only 2 nearest neighbors along the close-packed direction (the shortest lattice length), which doesn’t change when tilting one axis.

Simple monoclinic has CN= 2; however, there are no real materials that take this crystal structure.

Simple Monoclinic Lattice Constants

The monoclinic lattice has 4 parameters: lattice length a, b, and c; and lattice angle β. The other two lattice angles, α and γ, are always equal to 90º. There’s no real reason that the non-90º angle is defined as β; that’s just convention.

If we try to stay within the hard sphere model, we see that the close packed direction would be along one edge.

Presumably, this shortest lattice parameter would have a length of 2r because atoms touch along this side, and the other two lattice parameters would have a length greater than 2r. However, there would be no physical reason to have three lattice parameters of different length, which is why the simple monoclinic structure does not exist in real crystals.

If you wanted to describe the simple monoclinic crystal with math, you would describe the cell with the primitive vectors.

\vec{a}_{1}=a\hat{x}
\vec{a}_{2}=b\hat{y}
\vec{a}_{3}=c\cos{\beta}\hat{x}+c\sin{\beta}\hat{z}

Simple Monoclinic Atomic Packing Factor

There is no simple answer for the simple monoclinic APF, because the answer changes depending on the side lengths.

The volume of an monoclinic unit cell is abc\cdot\sin{\beta}, obtained by multiplying the square area of the base ab by the height of a cell. Since the c edge is NOT perpendicular to the base, we need to calculate the sine of the tilt angle (beta). Hence, the height of a cell equals \sin{\beta}.

There is 1 atom split between each of the 8 corners of the cell, and nothing in the center, so the total is one full atom with a spherical radius of \frac{4}{3}\pi r^3.

This allows us to write the theoretical APF in terms of the atomic radius and lattice constants.

APF=\frac{\frac{4}{3}\pi r^3}{abc\cdot\sin{\beta}}=\frac{4\pi r^}{3abc\cdot\sin{\beta}}

Final Thoughts

The simple monoclinic crystal system does NOT exist in real life, and only arises by mathematical definition. There are many crystals which have an monoclinic Bravais lattice, but these crystals have a more complicated “basis” which leads to its characteristic asymmetrical lattice parameters.

References and Further Reading

If you want to know more about the basics of crystallography, check out this article.

If you weren’t sure about the difference between crystal structure and Bravais lattice, check out this article.

I also mentioned atomic packing factor (APF) earlier in this article. This is an important concept in your introductory materials science class, so if you want a full explanation of APF, check out this page.

If you’re interested in advanced crystallography or crystallography databases, you may want to check out the AFLOW crystallographic library.

For a great reference for all crystal structures, check out “The AFLOW Library of Crystallographic Prototypes.”

The 14 Basic Crystal Structures

If you want to learn about specific crystal structures, here is a list of my articles about Bravais lattices and some related crystal structures for pure elements. Simple Monoclinic is one of these 14 Bravais lattices and also occurs as a crystal structure.

1. Simple Cubic
2. Face-Centered Cubic
2a. Diamond Cubic
3. Body-Centered Cubic
4. Simple Hexagonal
4a. Hexagonal Close-Packed
4b. Double Hexagonal Close-Packed (La-type)
5. Rhombohedral
5a. Rhombohedral Close-Packed (Sm-type)
6. Simple Tetragonal
7. Body-Centered Tetragonal
7a. Diamond Tetragonal (White Tin)
8. Simple Orthorhombic
9. Base-Centered Orthorhombic
10. Face-Centered Orthorhombic
11. Body-Centered Orthorhombic 
12. Simple Monoclinic 
13. Base-Centered Monoclinic 
14. Triclinic

Other articles in my crystallography series include:

What are Crystals and Grains
Introduction to Bravais Lattices
What is the Difference Between “Crystal Structure” and “Bravais Lattice”
Atomic Packing Factor
How to Read Miller Indices
How to Read Hexagonal Miller-Bravais Indices
Close-Packed Crystals and Stacking Order
Interstitial Sites
Primitive Cells
How to Read Crystallography Notation

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