quantum liquid crystal phases in strongly …sunkai/teaching/winter_2013/... · 2016-09-03 · •...
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Kai Sun
University of Michigan, Ann Arbor
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Course work and grading
Course grade will be based on • Problem set: 20% • Final Presentation: 20%
You will form teams of up to 3 members. Each team will give a 15 minute presentation on a topic you choose, illustrating an application of solid state physics.
• Midterm: 30% • Final: 30% • Piazza discussions: 5% bonus points
Homework will be assigned on Thursday and will be due the next Thursday Office hour: Tuesday or Wednesday (see the poll on CTools)
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How do we know there are atoms or molecules?
Brownian motion: random moving of particles suspended in a fluid
Now we have technique to “see” atom directly (Scanning tunneling microscope, developed in the 80s), but people knows the existences of atoms long before we can see them. Q: How did people know there are atoms before we can really see them? A1: For liquid/gas, from the study on Brownian motion (Einstein 1905) A2: For solids, from the X-ray crystallography (will be discussed next time)
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Matter
Matters
gas/liquid:
Atoms/molecules can move around
solids:
Atoms/molecules cannot move
Crystals:
Atoms/molecules form a periodic
structure
Random solids:
Atoms/molecules form a random
structure
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Salt
NaCl Na: Sodium (larger spheres) Cl: Chlorine (smaller spheres)
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An Ideal Crystal
An ideal crystal: infinite repetition of identical groups of atoms (e.g. NaCl). A group is called the basis (a unit cell) Q: How to describe and classify this periodic structure? A: Using Bravais lattices
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Bravais Lattices
Unit cell: each periodicity is called a unit cell. A unit cell may contain 1 or more atoms or molecules Bravais lattice: as easy as 1, 2, 3 1. Choose one point in each unit cell. 2. Make sure that we pick the same point in every unit cell. 3. These points form a lattice, which is known as the Bravais lattice. One can choose the point arbitrarily and different choices give the same Bravais lattice Many different materials can share the same Bravais lattice
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1D example
1D crystal 3 atoms/periodicity
Choice I:
Choice II:
Choice III:
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2D example
An example of 2D crystal (one atom per unit cell)
Choice I Choice II
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Another 2D example
An example of 2D crystal (two atoms per unit cell)
Choice I Choice II
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Another 2D example
An example of 2D crystal (three atoms per unit cell)
Choice I Choice II
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The translational symmetry of Bravais lattices
Symmetry: invariance under certain operation For Bravais lattices, translational symmetry is one of the most important. Translation: moving every point the same distance in the same direction, without rotation, reflection or change in size. Translational symmetry and translation vectors: If a system goes back to itself when we translate the system by some vector, we say that this system has a translational symmetry and this vector is known as a translation vector. There are an infinite number of translation vectors
If 𝑡 is a translation vector, n ∗ 𝑡 is also a translation vector where 𝑛 is an integer. For Bravais lattices, All Bravais lattices have translational symmetry. Any vector that connects two lattice points is a translational vector. For Bravais lattices, these translational vectors are also known as lattice vectors
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Interactive figures
http://www-personal.umich.edu/~sunkai/teaching/Winter_2013/translations.html
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Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)
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Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)
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Primitive lattice vectors
Q: How can we describe these lattice vectors (there are an infinite number of them)? A: Using primitive lattice vectors (there are only d of them in a d-dimensional space). For a 3D lattice, we can find three primitive lattice vectors (primitive translation vectors), such that any translation vector can be written as
𝑡 = 𝑛1𝑎 1 + 𝑛2𝑎 2 + 𝑛3𝑎 3 where 𝑛1, 𝑛2 and 𝑛3 are three integers. For a 2D lattice, we can find two primitive lattice vectors (primitive translation vectors), such that any translation vector can be written as
𝑡 = 𝑛1𝑎 1 + 𝑛2𝑎 2 where 𝑛1 and 𝑛2 are two integers. For a 1D lattice, we can find one primitive lattice vector (primitive translation vector), such that any translation vector can be written as
𝑡 = 𝑛1𝑎 1 where 𝑛1 is an integer.
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Primitive lattice vectors
Red (shorter) vectors: 𝑎 1 and 𝑎 2
Blue (longer) vectors: 𝑏1 and 𝑏2
𝑎 1 and 𝑎 2 are primitive lattice vectors
𝑏1 and 𝑏2 are NOT primitive lattice vectors
𝑏1 = 2𝑎 1 + 0 𝑎 2 𝑎 1 =1
2𝑏1 + 0𝑏2
Integer coefficients noninteger coefficients
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Primitive lattice vectors
The choices of primitive lattice vectors are NOT unique.
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Primitive cell in 2D
The Parallelegram defined by the two primitive lattice vectors are called a primitive cell.
the Area of a primitive cell: A = |𝑎 1 × 𝑎 2| Each primitive cell contains 1 site.
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Primitive cell in 3D
The parallelepiped defined by the three primitive lattice vectors are called a primitive cell.
the volume of a primitive cell: V = |𝑎 1. (𝑎 2 × 𝑎 3)| each primitive cell contains 1 site.
A special case: a cuboid
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Wigner–Seitz cell
the volume of a Wigner-Seitz cell is the same as a primitive cell each Wigner-Seitz cell contains 1 site (same as a primitive cell).
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Rotational symmetries
Rotational symmetries: If a system goes back to itself when we rotate it along certain axes by some angle 𝜃, we say that this system has a rotational symmetry. For the smallest 𝜃, 2𝜋/𝜃 is an integer, which we will call 𝑛. We say that the system has a 𝑛-fold rotational symmetry along this axis. For Bravais lattices, It can be proved that 𝑛 can only take the following values: 1, 2, 3, 4 or 6.
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Mirror planes
Mirror Planes:
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2D Bravais lattices
http://en.wikipedia.org/wiki/Bravais_lattice
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3D Bravais lattices
http://en.wikipedia.org/wiki/Bravais_lattice