Basics

To begin let’s just define some basic principals that are important to understand Solid State Physics.

The Reciprocal Lattice

We know that we can write the direct lattice as arbitrary combinations of the lattice vectors . We can calculate the reciprocal lattice vectors as follows

So that we can define the reciprocal lattice

The reciprocal lattice for common lattices we defined earlier can be seen in the following table

Direct Lattice	Reciprocal Lattice
SC (side )	SC (side )
FCC (side )	BCC (side )
BCC (side )	FCC (side )

Waves in Crystals

Now let’s consider a plane (e.g. electromagnetic) wave that interacts with a certain lattice. The wave will scatter at the atoms and we want to determine in which directions the outgoing waves constructively interfere with each other. Let’s start by looking at only two lattice points. We have an incoming wave with wave vector and the vector connecting the two lattice points is .

The wave’s path difference between the two lattice points is . We can calculate their length in relation to and the angles and .

The incoming wave vector is and the outgoing wave vector .

The length can be expressed as

and similarly

So to get constructive interference the sum of and has to be a multiple of the wavelength

To get the relation between the wave vector itself we multiply with and thus get

We have to consider the whole lattice so so that the conditions for all different must be met. These are the

As we discussed earlier the reciprocal lattice is given by

One can prove that all scattering vectors that are equal to a reciprocal lattice vector

fulfill the Laue conditions, which means

We showed earlier that

So we can conclude a relation between the physical distance between planes and the reciprocal lattice vector

This means is parallel to and perpendicular to the -plane.

We can describe the incoming wave with wavevector and angular frequency

The outgoing wave with wavevector and same frequency (we assume elastic scattering) is then

where are the scattering probabilities. It simplifies to

where

We call

the structure factor.

Sources

• https://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf
• https://courses.cit.cornell.edu/mse5470/handout5.pdf
• Introduction to Solid State Physics, Umeå Universitet, Sune Pettersson
• Experimentalphysik 3 Atome, Moleküle und Festkörper, Wolfgang Demtröder

Adiabatic process

During an adiabatic process there is by definition no heat exchanged with the environment. This can e.g. be approximately true for a very fast change in volume where there is almost no time to allow heat flow during this process. For us this is very convenient as we can now write the first law like this

The inner energy for an ideal gas is defined by

where is the degree of freedom for the gas, the number of gas particles, the Boltzmann constant and the temperature of the gas. As is the only variable that might change during the adiabatic process we can write the change of inner energy as

The work done on the gas/ on the environment is given by

where is the pressure and the change of volume during the process.

Combining those to the first equation we get

Now let’s introduce probably the most essential identity of Thermodynamics

We can now apply the differential operator to get and insert this into our equation

where one defines as the adiabatic index.

Let’s integrate this equation on both sides

Using and we get

Now by inserting we get

Example

As you may know, the Diesel Engine does not rely on spark ignition - the fuel can ignite without a spark needed. The fuel-air vapor is instead compressed so quickly that the temperature increases significantly - enough to start the power stroke.

E.g. The 2019 Ford Super Duty has a compression ratio of . We derived that and so

where is the temperature before the compression and the temperature after the compression (same for volume). Let’s assume that the incoming air is about and that we deal only with a diatomic gas with and therefore .

The final temperature can then be calculated with

Plugging in our values we get

Sources

• Inkscape Tutorial: Vector Flame Icon
• 2019 Ford Super Duty Specifications

Get Started

When trying to create your first commit git wants to know some info about you:

Author identity unknown

*** Please tell me who you are.

Run

  git config --global user.email "you@example.com"
  git config --global user.name "Your Name"

to set your account's default identity.
Omit --global to set the identity only in this repository.

So obviously follow the instructions and configure your email and name with:

git config --global user.email "you@example.com"
git config --global user.name "Your Name"

GPG key

First you can install the seahorse app, a password & encryption key manager. Using Arch Linux or Manjaro you can do that with:

sudo pacman -S seahorse

Then add a GPG key.

Enter a name, and the same email address that you use for your Github account.

After entering the passphrase for this key export the public key.

Copy the content of the exported file, sign in to your Github account and go to Settings->SSH and GPG keys->New GPG key and paste the public key there.

Now go back to seahorse and copy the key ID from the properties dialog of your GPG key (double click on the key). Then you need to tell Git about your GPG key with:

git config --global user.signingkey <KEY_ID>

(replace <KEY_ID> with your actual key ID)

Finished! 💪 🎊

From now on you can sign your commits with:

git commit -S -m "Hooray I can sign my commits"

SSH key

Using SSH you do not have to enter your token or password every time you push a commit to github. In seahorse click + then select Secure Shell key. Enter a brief description then click Create and set up and enter a passphrase. To get the public double click on the ssh key to get to the properties dialog. There is a Copy icon next to the Public Key label. After clicking it your public key is saved in your clipboard. All you have to do now is adding your SSH key to Github. Therefore sign in to your Github account and go to Settings->SSH and GPG keys->New SSH key. There you have a title and copy your public key in the box below.

Great! 🎉 If you want to force git to always use SSH instead of HTTPS you can do that with:

git config --global url."git@github.com:".insteadOf "https://github.com/"

I hoped this post helped you! Happy coding!

Sources

Features