Physics of Condensed Matter IPhysics of Condensed Matter I Faculty of Physics UW...
Transcript of Physics of Condensed Matter IPhysics of Condensed Matter I Faculty of Physics UW...
Born Oppenheimer approximation
Max Born
(1882-1970)
Jacob R. Oppenheimer
(1904-1967)
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Chemical bonding and molecules
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Chemical bonding and molecules
W.
Ibac
h
J. G
inte
r
Small distancebetween atomsbands (pasma)
Large distancebetween atomslevels (poziomy)
Dr hab. Darek Wasik2016-01-25 4
From the molecule to the solid state
J. G
inte
r
Small distancebetween atomsbands (pasma)
Large distancebetween atomslevels (poziomy)
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From the molecule to the solid state
Eg
Przerwa energetycznaEnergy gap
Hybridization
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Molecules
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Carbon
Hybridization
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Types of chemical bonds
http://oen.dydaktyka.agh.edu.pl/dydaktyka/chemia
Carbon
The binding energy per atom:C (diamond) 7.30 eVSi 4.64 eVGe 3.87 eV
Semiconductors
Be B C N O
Mg Al Si P S
Zn Ga Ge As Se
Cd In Sn Sb Te
II III IV V VI
Group IV: diamond, Si, GeGroup III-V: GaAs, AlAs, InSb, InAs...Group II-VI: ZnSe, CdTe, ZnO, SdS...
ionicity ionicity
Covalent
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Types of chemical bondsSemiconductors
Be B C N O
Mg Al Si P S
Zn Ga Ge As Se
Cd In Sn Sb Te
II III IV V VI
Group IV: diamond, Si, GeGroup III-V: GaAs, AlAs, InSb, InAs...Group II-VI: ZnSe, CdTe, ZnO, SdS...
ionicity ionicity
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Types of chemical bondsSemiconductors
Carriers: Impurities (domieszki):
holes
electrons
+
-
Acceptrs (p-type)
Donors (n-type)
Covalent bonding
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Types of chemical bonds
Covalent bonding
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Types of chemical bonds
Allotropes of carbon
Covalent bonding
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Types of chemical bonds
Gra
ph
ene
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Types of chemical bonds
Ibach. Luth
Covalent bonding (+ polar covalent)
Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7)
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Types of chemical bonds
Covalent bonding (+ polar covalent)
Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7)
Ionic bonding
Tablica 2.4. Values of electronegativity (according Pauling) for several major elements (for H set 2,1)
I II III IV V VI VII
Li1,0
Be1,5
B2,0
C2,5
N3,0
O3,5
F4,0
Na0,9
Mg1,2
Al1,5
Si1,8
P2,1
S2,5
Cl3,0
K0,8
Ca1,0
Ga1,6
Ge1,7
As2,0
Se2,4
Br2,8
Rb0,8
Sn1,7
J2,4
ionicity ionicity
Electronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc โฅ 1,7).
NaCl
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Types of chemical bonds
Ionic bondingElectronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc โฅ 1,7).
NaCl
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Types of chemical bonds
Convention:Covalent bond Dc โค 0,4
Polar Covalent 0,4 โค Dc โค 1,7
Ionic Bonds Dc โฅ 1,7
Ionic bondingElectronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc โฅ 1,7).
NaCl
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Types of chemical bonds
C. Kittel
The binding energy per pair of ions :NaCl 7.95 eVNaI 7.10 eVKBr 6.92 eV
Distribution of charge density in the NaCl
plane based on X-rayresults.
Ionic bondingElectronegativity (symb. c) - the tendency of an atom to attract electrons. In the extreme case when the electronegativity of both elements is very different (eg. Li and F), it comes to a full transfer of an electron toward more electronegative atom, which leads to the formation of ionic bond (Dc โฅ 1,7).
NaCl
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Types of chemical bonds
Ibach. Luth
๐ ิฆ๐๐๐ = ยฑ๐2
4๐๐0๐๐๐
Potential energy between two singly charged ions ๐, ๐
๐๐๐ = ๐ ๐๐๐
separation of nearest neighbours
depends on crystal structure
๐ ๐ =
๐โ ๐
โ๐2
4๐๐0๐
ยฑ1
๐๐๐= โ
๐2
4๐๐0๐
๐โ ๐
ยฑ1
๐๐๐= โ
๐2
4๐๐0๐๐ด
The Madelung constant ๐ด depends on the structure e.g. ๐ด๐๐๐ถ๐ = 1.748
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Covalent bonding (+ polar covalent)
Valence electrons are shared between atoms (non-polar Dc < 0,4; polar 0,4 < Dc < 1,7)
electrons are tranfered between atoms (Dc โฅ 1,7). An essential contribution to bonds energyof ionic crystals comes from the electrostatic interaction (Madelung energy):
๐ ๐ = ๐ โ๐2
4๐๐0๐
๐โ ๐
ยฑ1
๐๐๐+
๐ต
๐๐
๐โ ๐
1
๐๐๐๐
๐ โ the distance between atoms๐๐๐๐ - the distance between pair of ions ๐, ๐
๐ต, ๐ โ repulsive potential parameters (๐ = 6 โ 12)
๐ด = ฯ๐โ ๐ยฑ1
๐๐๐- the Madelung constant (for NaCl structure ๐ด = 1,748, for CsCl ๐ด = 1,763)
Ionic bonding
Types of chemical bonds
The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms.
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+
eโ
Na+ Na+ Na+ Na+
Na+ Na+ Na+ Na+ Na+
Na+ Na+ Na+ Na+ Na+
Na+ Na+ Na+ Na+ Na+
eโ eโeโ
eโeโ
eโeโ
eโeโ
eโeโ
eโeโ
eโeโ
eโ
eโ
Electron gas
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Types of chemical bondsMetalic bonding
The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms.
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Types of chemical bondsMetalic bonding
โข In the alkali metals only delocalized electrons of the last shell ๐๐ contribute to bonding. In these metals the length of the bonds can be easily changed (high compressibility)
โข The metals of further columns of the Periodic Table also deeper shells give an important contribution to bonding (in particular, transition metals and rare earths ๐ and ๐ shells). In these metals the length of the bonds is much harder to change (small compressibility)
โข The bonds in metals are usually not very strong, but there are also metals with quite strong bonding - eg. Tungsten (wolfram)
The chemical bond in metals, formed by the electrodynamic interaction between the positively charged atom cores, which are located in nodes of the lattice, and negatively charged plasma electrons (delocalized electrons, electron gas). Similar to a covalent bond, but electrons forming a bond are common to a large number of atoms.
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Types of chemical bondsMetalic bonding
Celulose
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Types of chemical bondsHydrogen bondingHydrogen is shared betweenatoms
Wiฤ zania
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C. Kittel
Hydrogen bondingHydrogen is shared betweenatoms
HF2โ
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Types of chemical bondsHydrogen bondingHydrogen is shared betweenatoms
http://www.smart-elements.com
Ne, Ar, Kr, Xe โ interaction of induced dipole moments.
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Types of chemical bondsVan der Waals bonds
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attractive forces between the positive end of one polar molecule and the negative end of another polar molecule - intermolecular interaction (e.g. ICl).
Types of chemical bondsDipole bonding (also intermolecular interaction)
https://saylordotorg.github.io/text_general-chemistry-principles-patterns-and-applications-v1.0/s15-02-intermolecular-forces.html
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Van der Waals bondsNe, Ar, Kr, Xe โ interaction of induced dipole moments.
Responsible for the possibility of condensation and solidification of noble gases (London ineraction)
โข interaction between permanent dipoles (Keesom interaction) โข interaction between permanent and induced dipoles (Debye interaction) โข London interaction โ London dispersion forces (interaction between induced dipoles)โข Lennard-Jones potential
๐ ๐ = 4๐๐
๐
12
โ๐
๐
6
๐๐ก๐๐ก ๐ = 2๐๐
๐โ ๐
๐
๐๐๐๐
12
โ
๐โ ๐
๐
๐๐๐๐
6
The potential energy of ๐ atoms
Types of chemical bondsVan der Waals bonds (also intermolecular interaction)
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Covalent bond Ionic bond Metallic bond
โข Directional bond(hybrydization)
โข Isolators orsemiconductors (chargebetween atoms)
โข Many of the covalent compounds dissolved in non-polar solvents, and areinsoluble in water
โข non-directional bondโข Isolators (charge in ions)โข Many of the ionic
compounds dissolved in a polar solvent (water) and not soluble in non-polar
โข non-directional bond, delocalised electrons
โข The more electrons, the stronger the bond
โข Conductors (free charge)โข Metals crystallize
preferentially in closedpacked structures (fcc, hcp, bcc)
โข Plastic (metal ions can easily move under the influence of an external force)
Types of chemical bonds
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Covalent bond Ionic bond Metallic bond
โข Directional bond(hybrydization)
โข Isolators orsemiconductors (chargebetween atoms)
โข Many of the covalent compounds dissolved in non-polar solvents, and areinsoluble in water
โข non-directional bondโข Isolators (charge in ions)โข Many of the ionic
compounds dissolved in a polar solvent (water) and not soluble in non-polar
โข non-directional bond, delocalised electrons
โข The more electrons, the stronger the bond
โข Conductors (free charge)โข Metals crystallize
preferentially in closedpacked structures (fcc, hcp, bcc)
โข Plastic (metal ions can easily move under the influence of an external force)
Types of chemical bonds
Crystals
Krysztaล
Ciaลo amorficzne
Lattice is a regular and periodic arrangement of points in space (lattice sites or lattice points). It is a mathematical abstraction; the crystal structure there is only when the base is uniquely assigned to each network node
primitive translation vectors
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Crystal structure
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
๐ ิฆ๐ = ๐ ิฆ๐ + ๐
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Primitive translation vectors are not selected unambiguously!
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Primitive translation vectors are not selected unambiguously!
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
primitive lattice cell
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Wigner-Seitzprimitive cell
C. Kittel
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
The basis (baza) may be a single atom, ion, a set of atoms, eg. proteins 105, positioned around each and every lattice point.
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
The basis (baza) may be a single atom, ion, a set of atoms, eg. proteins 105, positioned around each and every lattice point.
Basis
๐ 0๐
๐ ๐๐ = ๐ 0๐ + ๐
Crystals
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Crystal structure
j -j
A
AโBโ
Bt1C D
n t1 ๐ตโฒ๐ดโฒ = ๐ถ๐ท = ๐๐ก1 = ๐ก1 1 โ 2 cos๐
cos๐ =1
21 โ ๐
Crystals
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Crystal structure
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
High symmetry cell Primitive cell
Bravais lattice
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Crystal structure
Regularna
Tetragonalna
HeksagonalnaRombowa
Jednoskoลna
Trรณjskoลna
90
90
cba
cba
Auguste Bravais1811-1863
In three-dimensional space, there are 14 Bravais lattices.
They form 7 lattice systems
Romboedryczna
Bravais lattice
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Crystal structure
Regularna
Tetragonalna
Rombowa
JednoskoลnaRomboedryczna
Trรณjskoลna
90120
cba
90
cba
120
90
cba
90
cba
90
90
cba
cba
90
cbaIn three-dimensional space, there are 14 Bravais lattices.
They form 7 lattice systems
Heksagonalna
Bravais lattice
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Crystal structure
Bravais lattice
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Crystal structure
Example: close packed structure
1 layer A
Bravais lattice
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Crystal structure
Example: close packed structure
1 layer A
2 layer B
Bravais lattice
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Crystal structure
Example: close packed structure
1 layer A
2 layer B
3 layer A
B
A
B
Bravais lattice
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Crystal structure
Example: close packed structure
1 layer A
2 layer B
3 layer C
A
B
C
Bravais lattice
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Crystal structure
Example: close packed structure
fcc lattice
Hexagonal lattice with basis
hexagonal close-packed (HCP)
Bravais lattice
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Crystal structure
Example: close packed structure
fcc lattice
Hexagonal lattice with basis
hexagonal close-packed (HCP)
Bravais lattice
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Crystal structure
Example: close packed structure
hexagonal close-packed (HCP)
Hexagonal lattice with basis
Bravais lattice
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Crystal structure
Example: close packed structure
fcc lattice
Bravais lattice
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Crystal structure
Example: close packed structure
fcc lattice
Bravais lattice
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Crystal structure
Example: close packed structure
fcc lattice
Lattice points
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Crystal structure
001 011
010000
100
101 111
ยฝ ยฝ 1
ยฝ 0ยฝ
ยฝ ยฝ0
0ยฝ ยฝ
ยฝ1 ยฝ
1ยฝ ยฝ
110
fcc lattice
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Coordinates of a cell
Translation vectors of lattice
๐1 ิฆ๐1, ๐2 ิฆ๐2, ๐3 ิฆ๐3
Lattice points
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Crystal structure
001 011
010000
100
101 111
ยฝ ยฝ 1
ยฝ 0ยฝ
ยฝ ยฝ0
0ยฝ ยฝ
ยฝ1 ยฝ
1ยฝ ยฝ
110
fcc lattice
primitive translation vectors
๐ = ๐1 ิฆ๐ก1 + ๐2 ิฆ๐ก2 + ๐3 ิฆ๐ก3
Coordinates of a cell
Translation vectors of lattice
๐1 ิฆ๐1, ๐2 ิฆ๐2, ๐3 ิฆ๐3
๐1, ๐2, ๐3
Lattice directions
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Crystal structure
[001] 011
[010]000
[100]
[101] [111]
ยฝ ยฝ 1
ยฝ 0ยฝ
ยฝ ยฝ0
0ยฝ ยฝ
ยฝ1 ยฝ
1ยฝ ยฝ
[110]
[112]
fcc lattice
Negative values denoted by the bar over the number
๐ข าง๐ฃ ๐ค
0 0 1 0 0 เดค1
The set of any of integers relatively prime(coprime, wzglฤdnie pierwsze) which are to each other as the projections of a vector parallel to the direction of crystal axes..
Planes in the crystal
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Crystal structure
A family of lattice planes are written (โ๐๐), and denote the family of planes that intercepts the three points:
C
B
000
A
ิฆ๐1โ,ิฆ๐2๐,ิฆ๐3๐
If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity)
E.g.: A=2, B=3, C=6, plane (3,2,1)
โ๐๐ planeโ๐๐ set of planesโ๐๐ diectionsโ๐๐ set of directions
Also: the family of planes orthogonal to:โ ิฆ๐1 + ๐ ิฆ๐2 + ๐ ิฆ๐3
Where ิฆ๐1, ิฆ๐2, ิฆ๐3 are reciprocal lattice vectors
Planes in the crystal
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Crystal structure
A family of lattice planes are written (โ๐๐), and denote the family of planes that intercepts the three points:
ิฆ๐1โ,ิฆ๐2๐,ิฆ๐3๐
If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity)
E.g.: A=2, B=3, C=6, plane (3,2,1)
โ๐๐ planeโ๐๐ set of planesโ๐๐ diectionsโ๐๐ set of directions
Also: the family of planes orthogonal to:โ ิฆ๐1 + ๐ ิฆ๐2 + ๐ ิฆ๐3
Where ิฆ๐1, ิฆ๐2, ิฆ๐3 are reciprocal lattice vectors
C
B
000
A
[321]
Planes in the crystal
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Crystal structure
A family of lattice planes are written (โ๐๐), and denote the family of planes that intercepts the three points:
ิฆ๐1โ,ิฆ๐2๐,ิฆ๐3๐
If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity)
E.g.: A=2, B=3, C=6, plane (3,2,1)
โ๐๐ planeโ๐๐ set of planesโ๐๐ diectionsโ๐๐ set of directions
Also: the family of planes orthogonal to:โ ิฆ๐1 + ๐ ิฆ๐2 + ๐ ิฆ๐3
Where ิฆ๐1, ิฆ๐2, ิฆ๐3 are reciprocal lattice vectors
Planes in the crystal
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Crystal structure
A family of lattice planes are written (โ๐๐), and denote the family of planes that intercepts the three points:
ิฆ๐1โ,ิฆ๐2๐,ิฆ๐3๐
If one of the indices is zero, it means that the planes do not intersect that axis (1/0 = infinity)
(100) (110) (111)
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Planes in the crystal
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Crystal structure
(100) (110) (111)
(110) (120) (212)โ
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http://pl.wikipedia.org/wiki/Wskaลบniki_Millera
Planes in the crystal
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Crystal structure
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Planes in the crystal
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Crystalography
The crystalline structure is studied by means of the diffraction of photons, neutrons, electrons or other light particles
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