(2,0,0) plane for simple cubic lattice

[feiyangflash]

Hi All,

Does the plane of miller index (2,0,0) exist in simple cubic lattice?

It is easy to understand that (1,0,0) plane exist. But for (2,0,0) plane, I feel confused.

If (2,0,0) plane exists, the plane intersect with the x-axis at 1/2*a, if a is the lattice constant; and the plane is parallel with y-axis and z-axis. However, there is no atoms deployed in this plane! Is there any physical meaning for this plane, since no atom is there? When the X-ray diffraction is done, does this plane ever play a role?

Thank you!

Fei

 

[uby]

Hi Fei,

A simple cubic lattice can have a multiple-atom basis. For example, cesium chloride is simple cubic with a two atom basis. The atom at the center of the cell would create a diffraction at the (200) plane. In fact, for simple cubic, all reflections are possible due to the very high degree of symmetry.

 

[feiyangflash]

Hi UBY,

Thank you! Your explanation makes sense.

The alpha form of Polonium (Po) has a simple cubic crystal structure in a single atom basis. Does the plane (2,0,0) have a physical meaning for Polonium?

Hi Fei,

A simple cubic lattice can have a multiple-atom basis. For example, cesium chloride is simple cubic with a two atom basis. The atom at the center of the cell would create a diffraction at the (200) plane. In fact, for simple cubic, all reflections are possible due to the very high degree of symmetry.

 

[uby]

For a single-atom basis, a (200) reflection will still exist, but only as a family member of the (100) reflection. In other words, its intensity would be related to the intensity of the (100) reflection by a geometric factor indicating the fraction of planes present (1/2).

In 2D, draw a series of parallel lines and pretend they are the (100) planes. Now draw a series of dashed lines in between these. The dashed lines AND the parallel lines represent all the equivalent (200) reflections, but only half are actually present in the crystal (the dashed lines are unoccupied).

 

[alphy]

I can confirm that the (2,0,0) plane does indeed exist in a simple cubic lattice. While it may seem confusing at first that there are no atoms located within this plane, it is important to remember that the lattice points in a simple cubic lattice represent the centers of the atoms, not the actual positions of the atoms themselves. Therefore, the (2,0,0) plane does not necessarily need to have atoms located within it in order to exist in the lattice.

In terms of physical meaning, the (2,0,0) plane can still have an impact on the properties of the lattice. For example, it can affect the overall symmetry and diffraction patterns of the lattice. When X-ray diffraction is performed, the (2,0,0) plane may not directly contribute to the diffraction peaks, but it can still affect the overall diffraction pattern due to its presence in the lattice.

Overall, while it may seem counterintuitive that a plane with no atoms can exist in a lattice, it is a fundamental aspect of the structure and can still have an impact on its properties.