Dr hab. EWA POPKO popko [email protected] Room 231a, A-1 Modern Physics.
Lecture IX Crystals dr hab. Ewa Popko. The Schrödinger equation The hydrogen atom The potential...
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Transcript of Lecture IX Crystals dr hab. Ewa Popko. The Schrödinger equation The hydrogen atom The potential...
![Page 1: Lecture IX Crystals dr hab. Ewa Popko. The Schrödinger equation The hydrogen atom The potential energy in spherical coordinates (The potential energy.](https://reader035.fdocuments.in/reader035/viewer/2022062515/56649cdc5503460f949a7550/html5/thumbnails/1.jpg)
Lecture IX
Crystals
dr hab. Ewa Popko
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zyxEzyxzyxVzyxm
,,,,,,2 2
2
2
2
2
22
zyxEzyxH ,,,,ˆ
The Schrödinger equationThe hydrogen atom
The potential energy in
spherical coordinates
(The potential energy function is spherically symmetric.)
Partial differential equation with three independent variables
r
erV
2
04
1)(
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S-states probability
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P-states probability
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Why Solids?
most elements are solid at room temperature
atoms in ~fixed position
“simple” case - crystalline solid
Crystal Structure
Why study crystal structures?
description of solid
comparison with other similar materials - classification
correlation with physical properties
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Early ideas• Crystals are solid - but solids are not
necessarily crystalline• Crystals have symmetry (Kepler) and long
range order• Spheres and small shapes can be packed to
produce regular shapes (Hooke, Hauy)
?
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Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of
polygons in 2 dimensions, it can be shown why pentagons and heptagons shouldn’t occur.
Empty space not allowed
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CRYSTAL TYPES
Three types of solids, classified according to atomic arrangement: (a) crystalline and (b) amorphous materials are illustrated by microscopic views of the atoms, whereas (c) polycrystalline structure is illustrated by a more macroscopic view of adjacent single-crystalline regions, such as (a).
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quartz
Crystal structure
Amorphous structure
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Definitions1. The unit cell
“The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure”
The unit cell is a box with:
• 3 sides - a, b, c
• 3 angles - , ,
14 possible crystal structures (Bravais lattices)
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3D crystal lattice
cubica = b = c = =
tetragonala = b c = = = 90o
monoclinica b c = = 90o
90o
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orthorhombica b c = = = 90o
hexagonala = b c = = 90o; = 120o
triclinica b c 90o
trigonal (rhombohedral)a = b = c = = 90o
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Chemical bonding
Types:
Ionic bonding
Covalent bonding
Metallic bonding
Van der Walls bonding + -+ -
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Metallic bond
Atoms in group IA-IIB let electrons to roam ina crystal. Free electrons glue the crystal
Na+ Na+
e-
e-
Attract
Attract
Attract
AttractRepelRepel
Additional binding due to interaction of partially filled d – electron shells takes place in transitional metals: IIIB - VIIIB
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Core and Valence Electrons
Simple picture. Metal have CORE electrons that are bound to the nuclei, and VALENCE electrons that can move through the metal.
Most metals are formed from atoms with partially filled atomic orbitals.
e.g. Na, and Cu which have the electronic structure
Na 1s2 2s2 2p6 3s1
Cu 1s2 2s2 2p6 3s23p63d104s1
Insulators are formed from atoms with closed (totally filled) shells e.g. Solid inert gases
He 1s2 Ne 1s2 2s2 2p6
Or form close shells by covalent bonding i.e. Diamond
Note orbital filling in Cu does not follow normal rule
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sodium ion (Na+)
Ionic bonding• Metal atoms with 1 electron to lose can form
ionic bonds with non-metal atoms which need to gain 1 electron:– Eg. sodium reacts with fluorine to form sodium
fluoride:
sodium atom
(Na)
fluoride ion (F-)
fluorine atom
(F)
So the formula for
sodium fluoride is
NaF
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Examples of ionic bonding:NaCl•Each sodium atom is surrounded by its six nearest neighbor chlorine atoms (and vice versa)
•Electronically – sodium has one electron in its outer shell: [Ne]3s1 and Chlorine has 7 (out of 8 “available” electron positions filled in its outer shell) [Ne]3s23p5
•Sodium “gives up” one of its electrons to the chlorine atom to fill the shells resulting in [Ne] [Ar] cores with Na+ and Cl- ions
•Coulombic attraction with tightly bound electron cores
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Properties of the ionic crystals
• medium cohesive energy (2-4 eV/ atom).– low melting and boiling temp. .
• Low electrical conductivity.– (the lack of the free electrons).
• Transparent for VIS light– ( energy separation between neighbouring levels > 3 eV)
• Easily dissolved in water.– Electrical dipoles of water molecules attract the ions
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Covalent bonding: molecular orbitalsConsider an electron in the ground, 1s, state of a hydrogen atom
The Hamiltonian is
The expectation value of the electron energy is
This gives <E> = E1s = -13.6eV
o
2
4e = where
RadiusBohr theis a where a 1
= (r) i.e. oo e ar/-3/2 o
r
- 2m
- = H
22
(r)dV H (r) = > E <
+
E1s
V(r)
(r)
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Hydrogen Molecular Ion
Consider the H2+ molecular ion in which
one electron experiences the potential
of two protons. The Hamiltonian is
We approximate the electron wavefunctions as
and
|R - r|-
r -
2m
- = )rU( +
2m
- = H
2222
] + A[ |)] R - r(| + )r([ A = )r( 21
] B[ |)]R - r(| )r([ B = )r( 21
p+ p+
e-
R
r
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Bonding andanti-bonding states Expectation values of the energy are:
E = E1s – (R) for
E = E1s + (R) for
(R) - a positive function
Two atoms: original 1s stateleads to two allowed electron states in molecule.
Find for N atoms in a solid have N allowed energy states
)r(
)r(
)r(
-6 -4 -2 0 2 4 6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
r
-6 -4 -2 0 2 4 6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
r
V(r)
2)r(
)r(
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1s
2s
bonding
Anti-bonding
Anti-bonding
bonding
covalent bonding – H2 molecule
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• 8
6
4
2
0
-2
-4
-6
R00.1 0.2 0.3 0.4
nuclear separation (nm)
ener
gy(e
V)
parallel spin
antiparallel spin
system energy (H2)
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Covalent bonding
Atoms in group III, IV,V,&VI tend to form covalent bond
Filling factor
T. :0.34 F.C.C :0.74
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Covalent bonding
Crystals: C, Si, Ge
Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons areanti-parallel )
Example: Carbon 1S2 2S2 2p2
C C
Diamond: tetrahedron, cohesive energy 7.3eV
3D 2D
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Covalent Bonding in Silicon
•Silicon [Ne]3s23p2 has four electrons in its outermost shell
•Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice
•Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)
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Properties of the covalent crystals
• Strong, localized bonding.
• High cohesive energy (4-7 eV/atom).
– High melting and boiling temperature.
• Low conductivity.
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ionic – covalent mixed