Materia condensata

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Dispense delle lezioni di materia condensata del primo anno di magistrale della laurea in fisica alla Sapienza. Gli appunti sono in inglese, lingua originale del corso, e sono corredati con gli …

Esame Materia condensata

Facoltà Scienze matematiche fisiche e naturali

Dal corso del Prof. Caprara Sergio

Università Università degli Studi di Roma La Sapienza

A.A. 2018-2019

147 pagine

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APPUNTI DEL CORSO:

MATERIA CONDENSATA

ANNO 2018/2019

DOCENTE PROF. SERGIO CAPRARA

LINGUA INGLESE

APPUNTI DEL CORSO:

MATERIA CONDENSATA

ANNO 2018/2019

DOCENTE PROF SERGIO CAPRARA

LINGUA INGLESE

F-U-TS free energy of the system at low temperatures entropy gives low contribution to the free energy and the system tends to be ordinated. This is what we know as a crystal, an ordinated state where atoms and molecules occupy fixed positions.

An ideal crystal is infinite and the atoms that form it are fixed with a constant distance among them. The Bravais Lattice is a mathematical model for ideal crystals.

Suppose we have 3 different vectors, not in the same plane, called _₁, _₂, _₃, we can define _{R̅ = n_a_₁ + n_₂ + n_₃} where n ∈ N. The collection of all _R̅ is the Bravais Lattice.

In 1D there is just one B.L.

In 3D there are 4 crystal families that form 5 different B.L. All the B.L. have translational symmetry under the period of the lattice. There are also other symmetries: mirror symmetry, rotational symmetry, etc. Crystal families are listed from the the less symmetric to the most symmetric.

If _||≠|| and ≠ 90°, it has no other punctual symmetries (they leave one point invariant) than the inversion (if ∉ R̅ ∈ BL, so R̅ ∈ BL). This is called Monoclinic Lattice.

2)

All BL have inversion symmetry

2¹

|a| ≠ |b| θ = 90°

ORTHORHOMBIC

This kind of lattice, in addition to the symmetries we have already listed, has mirror symmetry.

3¹

|a| = |b| θ = 120°

HEXAGONAL

The resulting lattice is made of hexagons

We have mirror, rotation and rotation under 60° around one point symmetries

3)

In Landau’s book there is the proof that only quadratic and hexagonal shapes can form a lattice.

If we paint the same lattice than before but without the central point, this is not a BL because not all the points are equivalent: imagine to sit on the two points around you you don’t see the same things, so they are not equivalent (obv you have to go to a to another with a rigid translation.). So, not all the lattices are BL.

|ā| = |b̄| θ = 90°

TETRAGONAL

This is the most symmetric BL in 3D.

We can create a lattice in another way: covering the space with an unitary figure.

Each PRIMITIVE unite cell must contain just 1 lattice site. We can obtain the same BL using b̄' instead of b because b̄' is a linear combination of b̄.

There are some choices that make clear the symmetries of the lattice. Let's focus on the orthorhombic.

ā and b̄ are not primitive vectors because through a linear combination of them, we cannot reach the central point. ĉ and ā are the primitive vectors but they don't have the symmetries of the lattice.

CONVENTIONAL cell is not a primitive one but it has the same symmetries of the lattice.

It is always possible to have a primitive unite cell which is also conventional. It's called the WIGNER-SEITZ cell.

Th_5^th type of BL in 3D is the orthorhombic with a point in the center

It's the same BL.

In 3D we have 7 crystal families that form 14 BL.

TRICLINIC

|a|≠|b|≠|c| θ₁≠θ₂≠θ₃

It has only mirror symmetry. Triclinic is primitive.
MONOCLINIC

|a|≠|b|≠|c| θ₁=θ₃=90°, θ₃≠90°

Monoclinic is primitive. If we put a point in the center of one of the rectangular faces we obtain a BASE-CENTERED (Bc).

So we have 3 BL for the monoclinic family.

3^rd. ORTHOROMBIC

|ã| ≠ |b| ≠ |c| ₁ = ₂ = ₃ = 90°

Orthorombic is primitive, from this we can obtain a base centered; if we put a point in the center of all faces we have a FACE CENTERED; if we put a point in the middle of diagonals we obtain ← 4BL

4^th TETRAGONAL

|ã| = |b| ≠ |c| ₁ = ₂ = ₃ = 90°

We can only decorate it with a body-centered ← 3BL

6^th RHOMBOHEDRAL

|ã| = |b| = |c| ₁ = ₂ = ₃ ≠ 90° (it’s a cube compressed or stressed)

It’s only primitive. ← 1BL

5^th HEXAGONAL

We have an hexagonal in the plane and whatever we want in the height.

|ã| = |b| ₁ = 120 ₂ = ₃ = 90°

It’s only

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