Crystalline Structures

2024-12-07T02:52:19Z

MaxwellH: /* Point Defects: */

Crystalline structures are the foundation of solid-state materials, describing the orderly arrangement of atoms, ions, or molecules in three-dimensional space. These structures are characterized by their repeating patterns, known as lattices, which extend throughout the material. The study of crystalline structures is essential in understanding the physical and chemical properties of solids, including their strength, conductivity, thermal behavior, and optical properties.

==Bravais lattices==
At the heart of crystallography is the concept of the unit cell—the smallest repeating unit that defines the symmetry and geometry of the entire lattice. Unit cells can take on a variety of configurations, each classified under the Bravais lattices, which represent the 14 unique ways atoms can be arranged in space. We will only cover 3 of the main configurations.

[[File:CNX Chem 10 06 UnitCell1.jpg]]

The main types of configurations that a Bravais lattice can take are: Primitive, Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC). These configurations describe how atoms are arranged within a unit cell and are essential for understanding the structure and properties of crystalline materials. Among these, Primitive is the least efficient and weakest in terms of packing and structural strength, as its atoms are arranged only at the corners of the unit cell, leaving significant empty space. BCC, with an additional atom at the center of the unit cell, is stronger and has a higher packing efficiency than Primitive. However, FCC is the most efficient and stable configuration, with atoms occupying the corners and centers of all cube faces, resulting in the highest packing density and contributing to greater ductility and strength in materials that adopt this structure. These differences in packing and arrangement significantly influence the material properties, including density, strength, and thermal and electrical conductivity.

[[File:FCCBCC.png]]

====Primitive (Simple Cubic):====
In this configuration, atoms are located only at the corners of the unit cell. Each corner atom is shared with adjacent unit cells, resulting in one net atom per unit cell. The packing efficiency is low, making this arrangement relatively rare in nature.

Example: Polonium, Caesium Chloride

====Body-Centered Cubic====
In addition to the corner atoms, there is a single atom located at the center of the unit cell. This arrangement increases the packing efficiency compared to the primitive structure but still leaves some empty space within the lattice.

Example: Iron, Tungsten, Potassium (K)

====Face-Centered Cubic====
Atoms are present at each corner and the centers of all the cube faces. This configuration has the highest packing efficiency among the cubic structures (74%), leading to dense and stable crystal arrangements.

Example: Aluminum, Copper, Gold, Salt

==Impurities==
Impurities are atoms or molecules that are not part of the ideal crystal lattice but are present in the material. These can be either intentionally introduced or occur naturally during the crystal's formation. Impurities play a crucial role in determining the material's properties and are often exploited in applications like semiconductors and metallurgy.

===Types of Impurities===
====Substitutional Impurities:====
Foreign atoms replace the host atoms in the crystal lattice.

Example: In brass, zinc atoms substitute copper atoms.

====Interstitial Impurities:====
Smaller foreign atoms occupy the spaces (interstitial sites) between the host atoms in the lattice.

Example: Carbon atoms in the interstitial sites of iron, forming steel.

====Point Defects/Vacancy:====
Includes vacancies, where an atom is missing from the lattice, or foreign atoms that disrupt the crystal's regularity.

===Effects of Impurities===
Mechanical Properties: Impurities can strengthen a material by hindering dislocation movement (e.g., in alloys like steel).

Electrical Properties: In semiconductors, impurities (dopants) drastically alter conductivity. For example, adding boron to silicon creates a p-type semiconductor.

Optical Properties: Impurities can absorb or scatter light, influencing transparency or color.

===Applications===
Doping in Semiconductors: Controlled addition of impurities like phosphorus or boron to silicon enables the creation of Diodes or Transistors.

Alloying: Mixing metals with specific impurities enhances their strength, corrosion resistance, or workability such as Steel.

==Dislocations==

==Sources==
https://wisc.pb.unizin.org/minimisgenchem/chapter/types-of-unit-cells-body-centered-cubic-and-face-centered-cubic-m11q5/

https://wisc.pb.unizin.org/chem103and104/chapter/types-of-unit-cells-primitive-cubic-cell-m11q4/

Crystalline Structures

2024-12-07T02:50:53Z

MaxwellH: /* Types of Impurities */

Crystalline structures are the foundation of solid-state materials, describing the orderly arrangement of atoms, ions, or molecules in three-dimensional space. These structures are characterized by their repeating patterns, known as lattices, which extend throughout the material. The study of crystalline structures is essential in understanding the physical and chemical properties of solids, including their strength, conductivity, thermal behavior, and optical properties.

==Bravais lattices==
At the heart of crystallography is the concept of the unit cell—the smallest repeating unit that defines the symmetry and geometry of the entire lattice. Unit cells can take on a variety of configurations, each classified under the Bravais lattices, which represent the 14 unique ways atoms can be arranged in space. We will only cover 3 of the main configurations.

[[File:CNX Chem 10 06 UnitCell1.jpg]]

The main types of configurations that a Bravais lattice can take are: Primitive, Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC). These configurations describe how atoms are arranged within a unit cell and are essential for understanding the structure and properties of crystalline materials. Among these, Primitive is the least efficient and weakest in terms of packing and structural strength, as its atoms are arranged only at the corners of the unit cell, leaving significant empty space. BCC, with an additional atom at the center of the unit cell, is stronger and has a higher packing efficiency than Primitive. However, FCC is the most efficient and stable configuration, with atoms occupying the corners and centers of all cube faces, resulting in the highest packing density and contributing to greater ductility and strength in materials that adopt this structure. These differences in packing and arrangement significantly influence the material properties, including density, strength, and thermal and electrical conductivity.

[[File:FCCBCC.png]]

====Primitive (Simple Cubic):====
In this configuration, atoms are located only at the corners of the unit cell. Each corner atom is shared with adjacent unit cells, resulting in one net atom per unit cell. The packing efficiency is low, making this arrangement relatively rare in nature.

Example: Polonium, Caesium Chloride

====Body-Centered Cubic====
In addition to the corner atoms, there is a single atom located at the center of the unit cell. This arrangement increases the packing efficiency compared to the primitive structure but still leaves some empty space within the lattice.

Example: Iron, Tungsten, Potassium (K)

====Face-Centered Cubic====
Atoms are present at each corner and the centers of all the cube faces. This configuration has the highest packing efficiency among the cubic structures (74%), leading to dense and stable crystal arrangements.

Example: Aluminum, Copper, Gold, Salt

==Impurities==
Impurities are atoms or molecules that are not part of the ideal crystal lattice but are present in the material. These can be either intentionally introduced or occur naturally during the crystal's formation. Impurities play a crucial role in determining the material's properties and are often exploited in applications like semiconductors and metallurgy.

===Types of Impurities===
====Substitutional Impurities:====
Foreign atoms replace the host atoms in the crystal lattice.

Example: In brass, zinc atoms substitute copper atoms.

====Interstitial Impurities:====
Smaller foreign atoms occupy the spaces (interstitial sites) between the host atoms in the lattice.

Example: Carbon atoms in the interstitial sites of iron, forming steel.

====Point Defects:====
Includes vacancies, where an atom is missing from the lattice, or foreign atoms that disrupt the crystal's regularity.

===Effects of Impurities===
Mechanical Properties: Impurities can strengthen a material by hindering dislocation movement (e.g., in alloys like steel).

Electrical Properties: In semiconductors, impurities (dopants) drastically alter conductivity. For example, adding boron to silicon creates a p-type semiconductor.

Optical Properties: Impurities can absorb or scatter light, influencing transparency or color.

===Applications===
Doping in Semiconductors: Controlled addition of impurities like phosphorus or boron to silicon enables the creation of Diodes or Transistors.

Alloying: Mixing metals with specific impurities enhances their strength, corrosion resistance, or workability such as Steel.

==Dislocations==

==Sources==
https://wisc.pb.unizin.org/minimisgenchem/chapter/types-of-unit-cells-body-centered-cubic-and-face-centered-cubic-m11q5/

https://wisc.pb.unizin.org/chem103and104/chapter/types-of-unit-cells-primitive-cubic-cell-m11q4/

Crystalline Structures

2024-12-07T02:50:34Z

2024-12-07T02:14:48Z

MaxwellH: All Bravais lattices

== Summary ==
All Bravais lattices

2024-12-04T20:59:17Z

MaxwellH: Created blank page

Temperature & Entropy

2024-12-04T20:56:53Z

MaxwellH: /* Examples */

Claimed by Josh Brandt (April 19th, Spring 2022)

<blockquote>
<p>The increase of ... entropy is what distinguishes the past from the future, giving a direction to time.</p>
</blockquote>
<p>— Stephen Hawking</p>

Entropy is a fundamental characteristic of a system: highly related to the topic of Energy. It is often described as a quantitative measure of disorder. Its formal definition makes it a measure of how many ways there is to distribute energy into a system. What makes Entropy so important is its connection to the Second Law of Thermodynamics, which states

<blockquote>
<p>The entropy of a system never decreases spontaneously.</p>
</blockquote>

Entropy is highly related to [[Application of Statistics in Physics]], since it is a result of probability. The Second Law of Thermodynamics rests upon the fact that it is always more likely for entropy to increase, since it is more likely for an outcome to occur if there are more ways which it can. On macroscopic scales, the chances of entropy decrease as non-zero, but are so small as to never be observed.

The concept of Entropy is famous for its proof against the existence of Perpetual Motion Machines; disqualifying their ability to maintain order. The concept has a long and ambiguous history, but in recent science it has taken on a formal and integral definition.

==The Main Idea==

===A Mathematical Model===

The fundamental relationship between Temperature <math>T</math>, Energy <math>E</math> and Entropy <math> S \equiv k_B \ln\Omega</math> is <math>\frac{dS}{dE}=\frac{1}{T} </math>.

In order to understand and predict the behavior of solids, we can employ the Einstein Model. This simple model treats interatomic Coulombic force as a spring, justified by the often highly parabolic potential energy curve near the equilibrium in models of interatomic attraction (see Morse Potential, Buckingham Potential, and Lennard-Jones potential). In this way, one can imagine a cubic lattice of spring-interconnected atoms, with an imaginary cube centered around each one slicing each and every spring in half.

A quantum mechanical harmonic oscillator has quantized energy states, with one quanta being a unit of energy <math>q=\hbar \omega_0 </math>. These quanta can be distributed among a solid by going into anyone of the three half-springs attached to each atom. In fact, the number of ways to distribute <math>q</math> quanta of energy between <math>N</math> oscillators is <math>\Omega = \frac{(q+N-1)!}{q!(N-1)!} </math>. From this, it is clear that entropy is intrinsically related to the number of ways to distribute energy in a system.

From here, it is almost intuitive that systems should evolve to increased entropy over time. Following the fundamental postulate of thermodynamics: in the long-term steady state, all microstates have equal probability, we can see that a macrostate which includes more microstates is the most likely endpoint for a sufficiently time-evolved system. Having many microstates implies having many different ways to distribute energy, and thus high entropy. It should make sense that the Second Law of Thermodynamics states that "the entropy of an isolated system tends to increase through time".

The treatment of a large number of particle is the study of Statistical Mechanics, whose applications to Physics are discussed in another page ([[Application of Statistics in Physics]]). It is apparent that Thermodynamics and Statistical Mechanics have become linked as the atomic behavior behind thermodynamic properties was discovered.

===A Computational Model===

Entropy can be understood by watching the diffusion of gas in the following simulation:

https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html

This simulation connects two essential interpretations of Entropy. Firstly, notice that over time, although the gases begin separated, they end completely mixed. One could easily say that the disorder of the system increased over time, meaning the Entropy increased. Another way to look at it is from the perspective of statistics. There are more configurations where the particles can be disordered than ordered: there's only a couple ways all of the particles can be packed in to one corner, but uncountable ways they can be randomly dispersed. So left to its own, it is exceedingly likely to observe the particles reach a steady-state of perfect diffusion: where they are as mixed as possible. There's simply more ways for this to happen: it is the overwhelmingly likely outcome.

==Examples==

===Simple===

Question: Given that entropy is commonly applied to objects at everyday scales, explain why Boltzmann's definition of entropy is appropriate.

Solution: A pencil has a mass <math>\mathcal{O}(1 \text{ gram})</math>. Pencils are made of wood and graphite, so let's say a pencil is 100% Carbon. Carbon's molar mass is <math>12 \frac{\text{gram}}{\text{mole}} </math>. So there's <math>\frac{1}{12}</math> mole of Carbon atoms in a pencil. Thats <math>\frac{6 \times 10^{23}}{12}</math> Carbon atoms. For each atom, there are three oscillating springs under the Einstein Model, so there is <math>\mathcal{O}(10^{23})</math> oscillators, <math>N</math>. Obviously, <math>\Omega</math> will quickly become unmanageably large. Taking <math>\ln{10^{23}}</math> we get only <math>52</math>. To give appropriate units, we need a constant to multiply, which <math> k_B </math> gives. It should make sense why working with entropy on everyday scales is manageable under this definition.

===Middling===
Question: During a phase transition, a material's temperature does not increase: for example, a solid melting into a liquid or a liquid freezing into a solid. Explain how this is consistent with the equations of entropy.
[[File:Phases.jpeg]]

Solution: Entropy satisfies <math>\frac{dS}{dE} = \frac{1}{T} </math>. When a solid melts into a liquid, there is obviously energy flowing into it, and all things held constant the liquid has more entropy than the solid. So as energy is being dumped in during the transition, entropy is increasing, so <math>\frac{dS}{dE} > 0 </math>. But notice that this does not mean the temperature is increasing. <math>T</math> can be a constant while entropy changes with respect to energy. So this is consistent.

===Difficult===

Question: Show that there are <math>\Omega</math> ways to distribute <math>q</math> indistinguishable quanta between <math>N</math> oscillators where <math>\Omega = \frac{(N+q-1)!}{q!(N-1)!} </math>

Solution:
For illustration, let us first pick the <math>N=3</math> and <math>q=2</math> case and imagine the oscillators as buckets and the quanta as dots. The way we can distribute the quanta into the oscillators is shown below schematically

[[File:Quanta_buckes.jpeg]]

Instead of drawing it out, we can try and create a code that describes each configuration (triple bucket + dot set-up). Let's first imagine creating a "basis", something like this:

[[File:Basis.jpeg]]

With this basis, we could say that the above case 4 can be written as AB. Case 5 is BC and case 6 is AC. Keep in mind that since quanta are indistinguishable, AC and CA are equivalent and so are every other reordering. To designate two dots in one bucket, we can introduce a third letter, D. A D means to add another ball into the bucket in the combination with it, so case 1 is AD. All possible cases are AD, BD, CD, AB, BC, AC. Notice that these are all of the combinations of two letters from the set A,B,C,D.

Now let's take the more general case. If we have <math>N</math> buckets, we need at least <math>N</math> basis buckets. But, each bucket can have <math>q-1</math> extra dots inside, in addition to the single basis bucket. So, to represent all combinations, we need <math>N+q-1</math> "letters". In this scheme, we might say that each additional letter after the basis means to add an extra dot to the letter alphabetically the same with respect to its set to keep things unique. So if the basis is A,B,C then D means add an extra to A, and E would mean add an extra to C.

The amount of ways to distribute the dots is now <math>N+q-1 \choose q </math>, since we can pick sets of <math>q</math> letters from the <math>N+q-1</math> possibilities.

<math>{N+q-1 \choose q} = \frac{(N+q-1)!}{q!(N-1)!}=\Omega</math>

===MATLAB Code===

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% You may change one of these 2 variables, keep in mind once q is over 100

% It may be hard to compute

q = 10; % Quanta of energy

N = 3; % Oscillators

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

f = @(q,N)((factorial(q + (N - 1)))/(factorial(q) * factorial(N-1)));

y = [];

for i = 1:q

y1 = f(i,N);

y = [y y1];

end

x = [1:q];

plot(x,y);

xlim([1 q]);

ylim([0, y(end)]);

xlabel('q (Quanta of Energy)');

ylabel('\Omega (Multiplicity)');

title('Multiplicity vs. Energy Quanta');

box on

grid on;

==Connectedness==
#How is this topic connected to something that you are interested in?

I am very interested in astrophysics, and there are very interesting cosmological analyses of entropy. For example, why do planets and stars form if they're more ordered than the interstellar medium they come from? Is the entropy of the universe really increasing? These topics have implications far beyond classical scales.

#How is it connected to your major?

I'm a Physics major.

#Is there an interesting industrial application?

I'm a Physics major. But, I'll still answer: of course! You might say entropy is the opposing force of efficiency. In history, it was discovered in the context of heat loss with work by Carnot. If we could create infinite-machines, the world would have considerably less problems.

Entropy is highly related to the concept of [[Quantized energy levels]]. The idea that energy can only be distributed in quanta between atoms is essential to understanding the idea of how many ways you can distribute energy in a system. If energy was continuous inside of atoms, there would be infinite ways to distribute energy into any system with more than one "oscillator" in the Einstein Model.

==History==

The concepts of Temperature, Entropy and Energy have been linked throughout history. Historical notions of Heat described it as a particle, such as [[Sir Isaac Newton]] even believing it to have mass. In 1803, Lazare Carnot (father of [[Nicolas Leonard Sadi Carnot]]) formalized the idea that energy cannot be perfectly channeled: disorder is an intrinsic property of energy transformation. In the mid 1800s, [[Rudolf Clausius]] mathematically described a "transformation-content" of energy loss during any thermodynamic process. From the Greek word τροπή pronounced as "tropey" meaning "change", the prefix of "en"ergy was added onto the term when in 1865 entropy as we call it today was introduced. Clausius himself said <blockquote>I prefer going to the ancient languages for the names of important scientific quantities, so that they maymean the same thing in all living tongues. I propose, therefore, to call S the entropy of a body, after the Greek word "transformation". I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful.</blockquote> Two decades later, Ludwig Boltzmann established the connection between entropy and the number of states of a system, introducing the equation we use today and the famous Boltzmann Constant: the first major idea introduced to the modern field of statistical thermodynamics. [1]

==Notable Scientists==
*[[Josiah Willard Gibbs]]
*[[James Prescott Joule]]
*[[Nicolas Leonard Sadi Carnot]]
*[[Rudolf Clausius]]

== See also ==

All thermodynamics is linked to the concept of Energy. See also Heat, Temperature, Statistical Mechanics, Spontaneity. For Fun, see the [https://en.wikipedia.org/wiki/Heat_death_of_the_universe Heat Death of the Universe]

===Further reading===

*[[Application of Statistics in Physics]]
*[[Quantized energy levels]]
*[[Electronic Energy Levels and Photons]]

===External links===

*[https://en.wikipedia.org/wiki/Energy Energy]
*[https://en.wikipedia.org/wiki/Thermodynamic_equilibrium Thermodynamics]
*[https://en.wikipedia.org/wiki/Conservation_of_energy Conservation of Energy]
*[https://fs.blog/entropy/ Entropy]

==References==

1. “Entropy.” Wikipedia, Wikimedia Foundation, 23 Apr. 2022, https://en.wikipedia.org/wiki/Entropy

[[Category:Statistical Physics]]