Very simple vector question (solid state)

[Phys128]

Homework Statement

Graphene is a 2D crystal made entirely of carbon atoms. The crystal lattice can defined by the lattice vectors a and b, where the magnitudes |a|=|b|= 2.46 Å and the angle between the lattice vectors is 60°. The crystal has a basis of two carbon atoms, where the basis vectors are r1 = 0 and r2 = a/3 + b/3.

Sketch the unit cell of the crystal structure of graphene and calculate the carbon- carbon bond length (in Å).

Relevant Equations

Bond length = |r2|

Somewhat embarrassingly as a third year undergrad, this question has been completely stumping me for far too long now (2 hours). The solution is 1.42 Å and the working is given as

|r2| = 2cos(30)*1/3(2.46)

or alternatively

|r2| = (1/2|a|)/cos(30)

But I cannot grasp where this comes from. Please someone put me out of my misery

 

[ehild]

Problem Statement: Graphene is a 2D crystal made entirely of carbon atoms. The crystal lattice can defined by the lattice vectors a and b, where the magnitudes |a|=|b|= 2.46 Å and the angle between the lattice vectors is 60°. The crystal has a basis of two carbon atoms, where the basis vectors are r1 = 0 and r2 = a/3 + b/3.

Sketch the unit cell of the crystal structure of graphene and calculate the carbon- carbon bond length (in Å).
Relevant Equations: Bond length = |r2|

View attachment 246904

Somewhat embarrassingly as a third year undergrad, this question has been completely stumping me for far too long now (2 hours). The solution is 1.42 Å and the working is given as

|r2| = 2cos(30)*1/3(2.46)

or alternatively

|r2| = (1/2|a|)/cos(30)

But I cannot grasp where this comes from. Please someone put me out of my misery

You need the magnitude of the vector $\vec r_2$. How is the magnitude of a vector defined, using scalar product?

 

[Phys128]

You need the magnitude of the vector $\vec r_2$. How is the magnitude of a vector defined, using scalar product?

|r2| = |a||b|cosx but then should that not be [(2.46/3)^2]cos(30) rather than 2(2.46/3)cos30?

 

[ehild]

|r2| = |a||b|cosx but then should that not be [(2.46/3)^2]cos(30) rather than 2(2.46/3)cos30?

No, your formula would givse the scalar product of two vectors a and b, inclined at angle x.
The magnitude of a vector is defined as the square root of its scalar product by itself.
Or you can use the cosine rule to calculate the length of r2, but remember, the angle between a and b is 60°.

 

[TSny]

The comments of @ehild are great.

The following is not much different than her comments. But, maybe it will be helpful.

The solution is 1.42 Å and the working is given as

|r2| = 2cos(30)*1/3(2.46)

In order to see this, consider an isosceles triangle as shown:

Show that $z = 2x\cos\theta$. Apply this to your situation.

or alternatively

|r2| = (1/2|a|)/cos(30)

To see this way of expressing it, consider the right triangle shown in the figure below:

Show that the horizontal leg of the triangle has length $a/2$ as indicated. Once you have that, the result follows quickly.