<blockquote>Thermal Expansion of MgO</blockquote>This is the welcome page for the elective MgO component of the third year computational chemistry labs for 2018/19.

The computational lab will be from 10:00 to 17:00 on Monday, Tuesday, Thursday and Friday. During these 4 days demonstrators will be available in the computer room to answer all your questions.

'''There is an introductory lecture to the lab at 10:00 on the Start Date of your session. You will received and email with the location of this lecture.'''

The deadline is at 12:00 of the day indicated in the table below ("Report Deadline").

Work submitted late will be penalised according to the [https://www.imperial.ac.uk/media/imperial-college/administration-and-support-services/registry/academic-governance/public/academic-policy/marking-and-moderation/Late-submission-Policy.pdf Late Submission Policy].

There's a link set up on [http://bb.imperial.ac.uk Blackboard] for submitting the report in PDF and the Jupyter notebooks as a .zip folder in 3rd Year Chemistry Laboratories (2018 - 2019) / Y3C Third Year Computational Laboratory/.

In each exercise there are a number questions and sub-questions. The final report will be expected to contain answers to these questions.

Questions related to this computational experiment can be directed to the demonstrators, Dr Giuseppe Mallia and Prof. Nicholas Harrison.

== Introduction ==

The properties of materials (solids, liquids, gasses) are a statistical average over the many different energy states of the molecules making up the material. The vibrational free energy of H2 can be computed analytically by summing over the harmonic vibrations of the molecule. This cannot be done by hand for a real material containing many atoms.

In this laboratory you will use a simple model of atomic interactions to calculate the energy and vibrations of a crystal of MgO. These vibrational energy levels will then be used to compute the free energy of the crystal and to predict how the material expands when heated. In the last final stage you will go beyond the harmonic (and quasi-harmonic) approximation and expand the crystal using a technique called molecular dynamics - essentially reproducing the actual vibration motions of the atoms. Fortunately the computer will do most of the work !

== Instructions ==

- Download all the [https://github.com/imperialchem/MgO-lab/archive/master.zip Jupyter Notebooks as a .zip file].

- Open Jupyter Notebook.

- Run the The_lab.ipynb

- Follow the instructions in the notebook.

Please, make a copy of the Jupyter notebooks, just in case you mess up with the code.

== Previous years related contents ==

- Maths and Physics for Chemist

- Thermodynamics

- Statistical thermodynamics

- Python

Good skills on python is not a requirement. All the scripts are written in a way where you only need to write an input value (e.g. Temperature = 300 K). However, feel free to edit and play with the scripts.

== New contents ==

- Phonons and reciprocal space

- Quasi-Harmonic approximation

- Molecular dynamics

== Submission ==

The report will be written in a PDF document and submitted via Blackboard.

Additionally, all the files (including the lab notebooks) will be also submitted via Blackboard as a .zip folder.

=== Write up ===

The report structure will consist on three sections:
* Introduction/Summary (Half-page)
* Questions & answers (No page limit)
* Conclusions (Half-page)

Tips to write a report:
* The golden rule: Aim for clarity
** Structured statements that flow in a logical manner.
** Good use of diagrams and appropriate level of theory.
** Careful choice of content.

* Keep your language clear and simple.
* Label all tables and figures. Labels should be self-contained, which means that tables and figures should be interpretable by themself.
* Appropriate referencing of figures and tables.
* Cite previous works (with an accepted citation style) whenever is appropriate.

Introduction/Summary:
* The purpose of the Introduction/Summary is to put the reader in the context of the experiment and to explain how the experiment was carried in the lab. It may contain a brief review of previous research, why the research was undertaken, an explanation of the techniques and why they are used and why it is important in a broader context.

Questions & Answers:
* There are a number of questions in the lab script that has to be answered in this section of the report.
* Depending on the nature of the question, it might be appropriate to use figures or tables to give a proper answer.
* It is highly encourage to rationalise the answers.

Conclusions:
* The Conclusions gives a general description of the results and findings and it should be related back to the Introduction. If appropriate, suggest improvements or additional experiments.

=== Suggested Time Frame ===

Try to finish all the calculations by Thursday. All the calculations takes seconds, however, it takes time to analyse the results and understand all the new concepts that you will learn in this lab.

=== Mark Scheme ===

The break-down for the marks for this lab are as follows:

{|class=wikitable
|-
|Introduction/Summary
|20%
|-
|Questions & Answers
|60%
|-
|Conclusions
|20%
|}

=== Plagiarism ===

Submissions are checked for plagiarism. External images may be used if correctly cited, but it's always better to create your own.

== Demonstrators ==

The demonstrators will be Victor Chang (v.chang16@imperial.ac.uk) and Carles Rafols i Belles (c.rafols-i-belles16@imperial.ac.uk). They will be available in the computer room from 10:00 to 11:00 and from 2:00 to 3:00.

Feel free to contact them in the lab or via email.

== Related literature ==

[https://onlinelibrary.wiley.com/doi/epdf/10.1002/anie.198708461 How Chemistry and Physics meet in the Solid State by Roald Hoffman]

[https://www.neutron-sciences.org/articles/sfn/pdf/2011/01/sfn201112007.pdf Introduction to the theory of Lattice Dynamics]

Introduction to Lattice Dynamics. Dove, Martin T. (ebook available in the library)

ThirdYearMgOExpt-1415

2018-10-15T20:19:29Z

Vc2115: /* Installation of the lab in your personal computer */

ThirdYearMgOExpt-1415

2018-10-15T19:44:58Z

Vc2115: /* Installation of the lab in your personal computer */

2018-08-24T11:07:46Z

Vc2115:

== Vibrations ==

Recall that vibrations are dene as the oscillation (or movement) of atoms in a molecule (or crystal) in periodic motion. In the crystalline case, these periodic vibrations are known as phonons. The simplest approximation to describe this periodic motion (i.e. phonons) is that of the harmonic oscillator that is shown in Fig. 1 and described by Eq. 1.

{| class="wikitable"
![[File:Simple harmonic motion animation.gif | 300px]]
!<math> x(t) = Acos(\omega t + \phi) </math> (Eq. 1)
|-
!Fig. 1: Animation of a simple harmonic oscillator.
!<math>A</math> and <math>\phi</math> is the amplitud and the phase respectively and <math>\omega</math> is the angular frequency
|}

If we consider the bonds between atoms as elastic strings, we can consider the Hooke's Law (Eq. 2) as a good approximation.

<math> F= -k \Delta x \qquad E = \frac{kx^2}{2} \qquad k = \omega^2 m</math>

Here <math>k</math> denotes the spring constant and <math>m</math> is the mass.

== Vibrations in Solid State ==

In a simulation, we are first required to describe the system we are going to study. The "system" refers to the solid-state structure we want to investigate. Solid-state, by contrast to molecular science, makes use of the periodicity of a structure. Since the size of the crystal (or periodic tessellation of atoms) is significantly larger than the size of a single molecule in most cases, we assume that the number of atoms and structure of the crystal is infinite. Of course, not all solids have a perfect periodic arrangement and this makes things very complicated to simulate, however we will assume that our structure has a periodic arrangement.

As starting point, the simplest model where atoms are arranged periodically in space is the 1D chain of atoms with the same mass equally spaced by a (Notice that a is the equilibrium distance between atoms). Remember that in a vibration, the motion of the atoms can be described as a harmonic oscillator. Fig. 2 represent a 1D chain of atoms in a vibrational mode where the atoms are moving in anti-phase.

From Fig. 2 we note that when the atoms are far away from the equilibrium distance (Fig. 2 left and right), the energy will increase and when the distance between atoms is the same as the equilibrium distance (Fig. 2 centre), the energy is a minima. It is therefore possible to describe the change of the energy as a function of the position of the atoms in a vibrational mode with a function (Fig. 2 lower waves). To sum up, we have demonstrate that we only need a simple periodic function to describe the motion of the atoms in a crystal in a vibrational mode.

The next step is figuring out how many different vibrational modes are in a crystal. A simple rule to find the number of different vibrations is that the number of vibrational modes in a molecule is equal to 3N-6 (or 3N-5 if the molecule is linear) where N is the number of atoms. However, in a crystal we have an infinite number of atoms and therefore, we expected an infinite number of vibrational modes.
In Fig. 2 we have define a vibrational wave that corresponds to the moment where all the atoms are moving in antiphase. The opposite case, where all the atoms are moving in phase is shown in Fig. 3. When all the atoms are moving in phase, there is no change in the distance between atoms and the energy is constant. Since there is no change in energy (the frequency is equal to zero), the function that describes this vibration is a straight line.

We have defined two boundary conditions for the vibrational frequency. As we shown above, these correspond to the anti-phase movement of atoms (where the resultant vibrational energy is at a max value) and to in phase movement of atoms (which corresponds to a resultant vibrational frequency of zero). Therefore, it follows that if we have a maximum and a minimum frequency boundary, all the other frequencies in the crystal must lie in between these two (Fig. 4 shows an example of a vibration).

Once we have established a simple method to describe our system, we need to define a mathematical expression that describe the vibrations. Since the vibrational waves are also periodic functions, we can use a sine or cosine function (Eq. 3) to represent them.

This function is known as planewave, where A0 is the amplitude of the wave, k is the wavenumber, x is a given point in space, ! is the angular frequency, t is a given point in time and is the phase shift (which is considered 0 here). Remember that the wavenumber is defined as the number of wavelenghts per unit distance and k is just the number of radians per unit distance (Eq. 4).

We can also write this expression with a complex exponential form (Eq. 5).

Now we need a link between the function that describe the vibrations with the Hookes law (Eq. 2). By solving a series of equations it can be determined that the frequency of a particular vibration for a 1D chain of atoms is described by Eq. 6.

where J is the spring constant, M is the mass of the atom, a is the distance between atoms and k is the wavenumber (Eq. 4.)

Now, when the chain of atoms is made of two atoms with different masses, after solving again a set of equations we found that now the vibrational frequency is described by Eq. 7.

Notice that for a particular value of k, there are two different vibrations for the same wavenumber. This will be discuss later when we introduce the concept of phonon band structures.

== Reciprocal space ==

Here we are going to introduce the concept of reciprocal space (also known as k space). The reciprocal space is the Fourier transform of real space. This has used in crystallography and in fact, we have already applied this concept in the previous section. We found that we can describe the motion of a 1D chain of atoms with a single function. This function is a periodic (wave) function which is described by a wavenumber k. We demonstrate that all the vibrations are in between two states, when the motion of all the atoms are in phase and when the motion of all the atoms is in anti-phase. Notice that when all the atoms are moving in anti-phase, the wave that describes the vibration of the the 1D chain of atoms has a wavelength � that is equal to the equilibrium distance of the atoms a (Fig. 2).

In the opposite case, when the motion of all the atoms is in phase, we demonstrate that the function that describe the vibration is a constant function. In this case, we can consider that � is infinity and therefore, we need an infinite number of atoms to represent it (Fig. 3).

Therefore � goes from a to infinity. One can intuitively deduce that there is a smarter way to represent a vibration that requires a infinity and this is done by representing the inverse of infinity. Instead of defining� we can define the inverse of the the wavelength which is the wavenumbers k.

We know that k = 2� and the minimum value of � is a, therefore

For the opposite case, � ! 1

And we know that any other vibration is in between these two values. Therefore, we have define a lattice (reciprocal lattice) that goes from 0 to 2� a where we can study the vibrations (and any other property that is periodic with the distance of the atoms). Notice that for a 1D chain of atoms, we only need to define one atom and the equilibrium distance between atoms. This is known as the primitive cell (minimum unit cell). When we move to more complicated system like an actual crystal, we are dealing with 3 dimensions. In a crystal we need to define three lattice parameters and therefore we will have three reciprocal lattice. The space define by this three lattices is known as reciprocal space.

Phonons and reciprocal space

2018-08-24T11:06:24Z

2018-08-24T10:43:10Z

Vc2115: