Lecture 5 - Crystal geometry Flashcards
The lattice is a
geometric/algebraic concept correlated with the periodic translations of matter in the crystal
The lattice specifies the translational repetition within the crystal by
a set of points and/or the vectors that define the locations of those points
Each point in a lattice has
an identical environment with respect to the matter in the crystal and with respect to the other points of the lattice
The crystal systems classify crystals according to
the presence of particular symmetries within the crystal structure
The broadest system classification is the
crystal system
There are _____ standard 2D crystal systems
4
There are _____ standard 3D crystal systems
7
The number of crystal systems is equal to the number of
standard unit cell types
Oblique crystal system symmetry
1- or 2-fold rotations, no reflections
Rectangular crystal system symmetry
1- or 2-fold rotation + reflection symmetry
Square crystal system symmetry
4-fold rotation
Hexagonal crystal system symmetry
3- or 6-fold rotation
a, b, and y in oblique 2D unit cells
all unrestricted
a, b, and y in rectangular 2D unit cells
a and b unrestricted
y = 90 degrees
a, b, and y in square 2D unit cells
a = b
y = 90 degrees
a, b, and y in hexagonal 2D unit cells
a = b
y = 120 degrees
Filling the oblique unit lattice with a molecule creates an
oblique unit cell
Primitive unit cell
Lattice points only at the vertices
Centered unit cell
Lattice points at places other than vertices
Unit lattice + motif =
unit cell
Protein crystals are so fragile and sensitive to environmental changes because
only a few contacts exist within the crystal
Only ______ and ______ unit cells allow an entirely arbitrary choice of origin
primitive p1 and P1
Any crystallographic symmetry operation must generate
an identical copy of the motif
Translational restrictions limit all crystallographic rotation operations to
2-, 3-, 4-, and 6-fold rotations
Unit cell
The translationally repeated motif that is linked to a repeated volume
Asymmetric unit
A smaller box in the unit cell that has internal symmetry and contains the truly unique atoms
The asymmetric unit of a unit cell contains all the necessary information to generate the
complete unit cell of a crystal structure by applying its symmetry operations to the asymmetric unit
Translation of the molecules related by 2-fold axis generates
additional 2-fold symmetry axes
The tetragonal unit cell is generated by rotation around a
4-fold axis
Translating the p4 plane structure creates new
2-fold and 4-fold axes
The asymmetric unit of p4 covers _____ of the unit cell
1/4
A hexagonal tile can be divided into ______ equivalent rhomboids
three
A hexagon can be created from
three trigonal unit cells (rotated 120) or hexagonal unit cells (rotated 60)
Hexagonal internal symmetry creates additional 2-fold axes on
the cell edges and in the center of the hexagonal unit cell
In a trigonal p3 structure, after generating the unit cell contents by a 3-fold rotation, lattice translations a and b generate
the structure (2-D crystal)
For small molecules, there are _____ plane groups
17
For macromolecules, there are _____ plane groups
5 (p1, p2, p3, p4, p6)
In proteins, there are no mirror planes, only
translations and screw axes
Two-fold screw axis (21)
2-fold rotation, followed by translation b/2 parallel to b
Three-fold screw axis (31)
Rotation of 120 degrees followed by translation z=1/3 parallel to c
Three-fold screw axis (32)
Rotation of 120 degrees followed by translation z=2/3 parallel to c
The symbol for a screw axis is
nm (m is a subscript)
For a screw axis, n idicates
the type of rotation
For a screw axis, the translation is ____ of the unit cell
m/n
Non-crystallographic symmetry (NCS) exists when
more than one ‘identical’ object is present in the asymmetric unit
Each point (p) in a 3-D lattice can be assigned a unique
real space lattice vector (r)
The components of ‘r’ are given in
fractions of the unit cell vectors a, b, c
x, y, and z are dimensionless crystallographic coordinates called
fractional coordinates
For any standard assignment of the lattice and unit cell, lattice points occur at
all vertices of the cell
A primitive unit cell has lattice points only at the vertices of the cell and contains one copy of
the crystal’s translational motif
Can a primitive cell be assigned to every crystal?
Yes
A non-primitive cell contains ______ copy of the translationally replicated motif, and ______ lattice point
more than one; more than one
Non-primitive unit cells have additional lattice points at
locations other than the vertices
Non-primitive lattices/cells are preferred for particular structures because
they correlate the basis vectors of the unit cell with directions of symmetry elements
Six primitive 3-D lattices
Triclinic
Monoclinic
Orthorhombic
Trigonal, hexagonal
Tetragonal
Cubic
Symbols associated with 3D Bravais lattices are show as
italic capital letters
Bravais lattice symbol P
primitive
Bravais lattice symbol A, B, C
face-centered (on A, B, or C)
Bravais lattice symbol I (eye)
body centered
Bravais lattice symbol F
centered on all faces
Bravais lattice symbol R
rhombohedral
8 centered Bravais lattices
Monoclinic (C)
Tetragonal (I)
Orthorhombic (I, F, C)
Cubic (I, F)
Trigonal (R)
How many space groups are there for macromolecules?
65
Most common space group for macromolecules
P2(1)2(1)2(1)
If A is an m x r matrix, and B is an r x n matrix, the product AB is
m x n
2-fold rotation symbolic operators
-x, -y, z
2-fold screw symbolic operators
-x, -y, z + 1/2
For the matrix W(R,T), W is the
symmetry operator
For the matrix W(R,T), R is the
rotational matrix
For the matrix W(R,T), T is the
translational component (column vector)
Each symmetry operation W can be expressed as a
3 x 3 rotation matrix, R, and a translational component T
Space groups are mathematical groups of
operators (specifically symmetry operators)
The conditions for forming a space group G with elements g1, 2, …j are
closure
identity
inversion
associativity
Space groups with freely selectable origins are called
polar space groups
Any point with fractional coordinate vector x can be converted into the Cartesian coordinate vector X by
multiplication with the orthogonalization matrix O
The relationship between the real space and reciprocal space is a
Fourier transform
Where is real space?
The crystal
Where is reciprocal space?
The diffraction pattern
The reciprocal lattice is
a mathematical construct that simplifies metric calculations
A lattice plane of given Bravais lattices is a
a plane whose intersections with the lattice are periodic and intersect the Bravais lattice
A lattice plane is any plane containing
at least three noncolinear Bravais lattice points
All lattice planes can be described by a set of integer
Miller indices
As h and k increase, the corresponding interplanar distance, d,
decreases (small d)
In the case of an orthogonal real lattice, the direction of each reciprocal axis coincides with
the direction of its corresponding real axis
Following the construction rules, we can extend the reciprocal lattice to
fill the reciprocal space
Each and every lattice vector, rhk, corresponds to
a distinct set of lattice planes, hk
The tighter the lattice plane spacing, the
larger the extent of the reciprocal space becomes
An important direct consequence of the translation symmetry of lattices is that the
indices of centrosymmetrically related lattice planes are also centrosymmetric
The centrosymmetrically related lattice planes hk and -h-k generate reciprocal lattices vectors pointing in
opposite directions and of equal magnitude
a* is normal to plane
b,c
Sets of parallel and equidistant lattice planes are defined by their
Miller indices, hkl
The Miller indices are integer numbers indicating the number of
intercepts of a set of lattice planes with each of the unit cell axes
The closer the interplanar spacing, the ______ the indices hkl
larger
The larger the indices hkl, the ______ the planes hkl “slice” or sample the cell
more tightly
Tight sampling means ______ information
more
High hkl provide ______ detail about the sampled structure
more
