What Are the Dimensions of a 1x1 Unit Cell on a Cu(110) Surface?

[Eduard1]

Hi All,

Let's consider a Copper bulk material. This one can be reproduce (ad infinitum) by using a cubic unit cell (fcc) of lattice constant a. Let's cut this bulk along the (110) plane and expose the Cu(110) surface to the vacuum. My question is: which are the x and y dimensions of a (1x1) unit cell ? After my calculations the 1x1 unit cell, on the Cu(110) surface, will have the x and y dimensions of a and a*sqrt(2). However, according to my solid state course, the x and y dimensions of a 1x1 unit cell, on the Cu(110) surface, are of a and a*sqrt(2)/2. I simply can not understand from where it is coming the division to 2. I would expect to have only a*sqrt(2) as this is a cube face diagonal (and we cut along a cube face diagonal to get the 110 surface).

Can anyone help me with an explanation?

With all the best wishes,
Eduard

 

[BigJDubs]

Hi Eduard,The length of the diagonal is indeed a*sqrt(2). However, the (1x1) unit cell on the Cu(110) surface consists of two atoms, which are separated by a distance of a*sqrt(2)/2. The length of the diagonal is thus divided by two as it is the distance between the two atoms. Thus, the x and y dimensions of a (1x1) unit cell on the Cu(110) surface will be a and a*sqrt(2)/2. Hope this helps!

 

[charng]

Hello Eduard,

Thank you for your question. The reason for the division of a*sqrt(2) by 2 in the x and y dimensions of a (1x1) unit cell on the Cu(110) surface is due to the orientation of the atoms in the unit cell. In a cubic unit cell, the atoms are arranged in a face-centered cubic (fcc) structure, with atoms at each corner and in the center of each face. When the bulk material is cut along the (110) plane, the atoms on the surface are no longer arranged in a fcc structure, but rather in a different arrangement known as a square lattice. This square lattice has a different spacing between atoms compared to the fcc structure, resulting in a division by 2 in the unit cell dimensions. This can also be seen by considering the lattice parameters of the fcc and square lattice structures, where the latter has a spacing of a*sqrt(2)/2 between atoms along the (110) direction. I hope this explanation helps clarify the difference in unit cell dimensions between the bulk material and the surface.

Best regards,