# Very simple vector question (solid state)

• Phys128
In summary, the question is about finding the carbon-carbon bond length in graphene, a 2D crystal made of carbon atoms. Using the lattice vectors and basis vectors provided, the solution is found to be 1.42 Å. This can be calculated by taking the magnitude of vector r2, which can be found using the scalar product or the cosine rule. The solution can also be derived using the trigonometric relationship between the sides of an isosceles triangle and a right triangle.
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

Phys128 said:
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?

ehild said:
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?

Phys128 said:
|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°.

The comments of @ehild are great.

The following is not much different than her comments. But, maybe it will be helpful.
Phys128 said:
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.

## 1. What is a vector in solid state physics?

A vector in solid state physics is a quantity that has both magnitude and direction and can be described using mathematical coordinates. In solid state physics, vectors are often used to represent physical properties such as velocity, force, and electric/magnetic fields.

## 2. How is a vector represented in solid state physics?

In solid state physics, a vector is typically represented using an arrow pointing in the direction of the vector with its length proportional to the magnitude of the vector. Alternatively, vectors can also be represented using mathematical notation, such as <x,y,z> for a three-dimensional vector.

## 3. What are some common vector operations in solid state physics?

Some common vector operations in solid state physics include addition, subtraction, multiplication by a scalar, and dot/cross products. These operations allow for the manipulation and analysis of vectors in various physical systems.

## 4. How are vectors used in the study of solid state materials?

Vectors are used extensively in the study of solid state materials to describe and analyze various physical properties such as crystal structure, electronic band structure, and magnetic ordering. Vector analysis allows for a deeper understanding of these properties and how they contribute to the behavior of solid state materials.

## 5. Can vectors be used to describe the motion of particles in solid state materials?

Yes, vectors can be used to describe the motion of particles in solid state materials. This is often done using the concept of a wavevector, which represents the velocity and direction of a particle's motion in a material. Vectors are also used to describe the motion of electrons in a solid, which is essential in understanding electronic properties of materials.

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