crystallography ( 晶体学 ) crystallography ( 晶体学 ). structure is important type of...
TRANSCRIPT
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crystallogra
phy( 晶体学 )
crystallogra
phy( 晶体学 )
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Structure is important
Type of structure we discussed called
crystal structure ( 晶体结构 )
In crystals, atom groups (unit cells) are repeated
to form a solid material
Structure is important
Type of structure we discussed called
crystal structure ( 晶体结构 )
In crystals, atom groups (unit cells) are repeated
to form a solid material
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Structure is important
"X-ray diffraction" is important methodfor study of materials
Structure is important
"X-ray diffraction" is important methodfor study of materials
X-ray diffraction works because of the repeating nature of crystals
To understand X-ray diffraction, must firstunderstand repetition in crystals
The study of repetition in crystals is called crystallography
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Repetition = symmetry (对称性 )
Types of repetition:
Rotation (旋转 ) Translation (平移 )
Repetition = symmetry (对称性 )
Types of repetition:
Rotation (旋转 ) Translation (平移 )
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Rotation
What is rotational symmetry?
Rotation
What is rotational symmetry?
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Imagine that this object will be rotated (maybe)Imagine that this object will be rotated (maybe)
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Imagine that this object will be rotated (maybe)Imagine that this object will be rotated (maybe)
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Imagine that this object will be rotated (maybe)Imagine that this object will be rotated (maybe)
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Imagine that this object will be rotated (maybe)Imagine that this object will be rotated (maybe)
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Was it?Was it?
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The object is obviously symmetric…it has symmetryThe object is obviously symmetric…it has symmetry
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The object is obviously symmetric…it has symmetry
Can be rotated 90° w/o detection
The object is obviously symmetric…it has symmetry
Can be rotated 90° w/o detection
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…………so symmetry is really
doing nothing…………so symmetry is really
doing nothing
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Symmetry is doing nothing - or at least doing something so that it looks like nothing was done!
Symmetry is doing nothing - or at least doing something so that it looks like nothing was done!
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What kind of symmetry does this object have?What kind of symmetry does this object have?
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What kind of symmetry does this object have?What kind of symmetry does this object have?
4 ( 旋转轴 )4 ( 旋转轴 )
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What kind of symmetry does this object have?What kind of symmetry does this object have?
44
m ( 镜 )m ( 镜 )
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What kind of symmetry does this object have?What kind of symmetry does this object have?
44
mm
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What kind of symmetry does this object have?What kind of symmetry does this object have?
44
mm
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What kind of symmetry does this object have?What kind of symmetry does this object have?
44
mm
4mm( 点群 )4mm( 点群 )
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Another example:Another example:
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Another example:Another example:
66
mm
6mm6mm
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And another:And another:
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And another:And another:
22
22
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What about translation?
Same as rotation
What about translation?
Same as rotation
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What about translation?
Same as rotation
Ex: one dimensional array of points
What about translation?
Same as rotation
Ex: one dimensional array of points
Translations are restricted to only certain values toget symmetry (periodicity)Translations are restricted to only certain values toget symmetry (periodicity)
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2D translations
Lots of common examples
2D translations
Lots of common examples
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Each block is represented by a pointEach block is represented by a point
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This array of points is a LATTICE (晶格 )This array of points is a LATTICE (晶格 )
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Lattice - infinite ( 无限的 ), perfectly
periodic ( 周期性的 ) array of points in
a space
Lattice - infinite ( 无限的 ), perfectly
periodic ( 周期性的 ) array of points in
a space
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Not a lattice:Not a lattice:
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Not a lattice:Not a lattice:
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Not a lattice - ….some kind of STRUCTURE because not just pointsNot a lattice - ….some kind of STRUCTURE because not just points
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Another type of lattice - with a different symmetry
rectangular
Another type of lattice - with a different symmetry
rectangular
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Another type of lattice - with a different symmetry
square
Another type of lattice - with a different symmetry
square
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Another type of lattice - with a different symmetry
hexagonal
Another type of lattice - with a different symmetry
hexagonal
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Back to rotation - This lattice exhibits 6-fold symmetry
hexagonal
Back to rotation - This lattice exhibits 6-fold symmetry
hexagonal
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Periodicity and rotational symmetry
What types of rotational symmetry allowed?
Periodicity and rotational symmetry
What types of rotational symmetry allowed?
object with 4-fold symmetry translates OKobject with 4-fold symmetry translates OK
overlap not allowedoverlap not allowed
object with 5-fold symmetry doesn't translateobject with 5-fold symmetry doesn't translate
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Periodicity and rotational symmetry
Suppose periodic row of points is rotated through ± :
Periodicity and rotational symmetry
Suppose periodic row of points is rotated through ± :
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Periodicity and rotational symmetry
To maintain periodicity:
Periodicity and rotational symmetry
To maintain periodicity:
vector S = an integer x basis translation tvector S = an integer x basis translation t
S
tt
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vector S = an integer x basis translation tvector S = an integer x basis translation tt cos = S/2 = mt/2t cos = S/2 = mt/2
m cos axis
2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4-1 -1/2 2π/3 4π/3 3-2 -1 - π π 2
m cos axis
2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4-1 -1/2 2π/3 4π/3 3-2 -1 - π π 2
S
tt
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Only rotation axes consistent with lattice periodicity in 2-D or 3-DOnly rotation axes consistent with lattice periodicity in 2-D or 3-D
m cos axis
2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4-1 -1/2 2π/3 4π/3 3-2 -1 - π π 2
m cos axis
2 1 0 2 π 1 1 1/2 π/3 5π/3 6 0 0 π/2 3π/2 4-1 -1/2 2π/3 4π/3 3-2 -1 - π π 2
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We abstracted points from the shape:We abstracted points from the shape:
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We abstracted points from the shape:We abstracted points from the shape:
Now we abstract further:Now we abstract further:
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Now we abstract further:Now we abstract further:
This is a UNIT CELLThis is a UNIT CELL
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Now we abstract further:
This is a UNIT CELL
Represented by two lengths and an angle
…….or, alternatively, by two vectors
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Basis vectors and unit cellsBasis vectors and unit cells
T = t a + t b
a b
a and b are the basis vectors for the lattice
ab
T
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In 3-D:
a
b
a, b, and c are the basis vectors for the lattice
c
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In 3-D:
a, b, and c are the basis vectors for the lattice
T = t a + t b + t ca b c
a
b
c
–––> [221] direction
T
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aa
bb
cc
Lattice parameters ( 晶格参数 ) :Lattice parameters ( 晶格参数 ) :
need 3 lengths - |a|, |b|, |c| & 3 angles - , , to get cell shape
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The many thousands of lattices classified into crystal systems ( 晶系 )
The many thousands of lattices classified into crystal systems ( 晶系 )
System Interaxial Axes
Angles
Triclinic ≠ ≠ ≠ 90° a ≠ b ≠ cMonoclinic = = 90° ≠ a ≠ b ≠ cOrthorhombic = = = 90° a ≠ b ≠ cTetragonal = = = 90° a = b ≠ cCubic = = = 90° a = b = cHexagonal = = 90°, = 120° a = b ≠ cTrigonal = = 90°, = 120° a = b ≠ c
System Interaxial Axes
Angles
Triclinic ≠ ≠ ≠ 90° a ≠ b ≠ cMonoclinic = = 90° ≠ a ≠ b ≠ cOrthorhombic = = = 90° a ≠ b ≠ cTetragonal = = = 90° a = b ≠ cCubic = = = 90° a = b = cHexagonal = = 90°, = 120° a = b ≠ cTrigonal = = 90°, = 120° a = b ≠ c
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The many thousands of lattices classified into crystal systemsThe many thousands of lattices classified into crystal systems
System Minimum symmetry
Triclinic 1 or 1Monoclinic 2 or 2Orthorhombic three 2s or 2sTetragonal 4 or 4Cubic four 3s or 3s
Hexagonal 6 or 6Trigonal 3 or 3
System Minimum symmetry
Triclinic 1 or 1Monoclinic 2 or 2Orthorhombic three 2s or 2sTetragonal 4 or 4Cubic four 3s or 3s
Hexagonal 6 or 6Trigonal 3 or 3
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For given lattice, infinite number ofunit cells possible:For given lattice, infinite number ofunit cells possible:
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When choosing unit cell, pick:
Simplest, smallest Right angles, if possible Cell shape consistent with symmetry
When choosing unit cell, pick:
Simplest, smallest Right angles, if possible Cell shape consistent with symmetry
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Which one? Why?Which one? Why?
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Within each crystal system, different types ofcentering consistent with symmetryWithin each crystal system, different types ofcentering consistent with symmetry
System Allowed centering
Triclinic P (primitive) Monoclinic P, I (innerzentiert)
Orthorhombic P, I, F (flächenzentiert), A (end centered) Tetragonal P, I Cubic P, I, F Hexagonal P Trigonal P, R (rhombohedral centered) (P hex = P trig)
The 14 Bravais lattices
System Allowed centering
Triclinic P (primitive) Monoclinic P, I (innerzentiert)
Orthorhombic P, I, F (flächenzentiert), A (end centered) Tetragonal P, I Cubic P, I, F Hexagonal P Trigonal P, R (rhombohedral centered) (P hex = P trig)
The 14 Bravais lattices