Lecture 5 - Crystal geometry Flashcards by Grant Meyer

The lattice is a

geometric/algebraic concept correlated with the periodic translations of matter in the crystal

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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

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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

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The crystal systems classify crystals according to

the presence of particular symmetries within the crystal structure

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The broadest system classification is the

crystal system

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There are _____ standard 2D crystal systems

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There are _____ standard 3D crystal systems

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The number of crystal systems is equal to the number of

standard unit cell types

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Oblique crystal system symmetry

1- or 2-fold rotations, no reflections

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Rectangular crystal system symmetry

1- or 2-fold rotation + reflection symmetry

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Square crystal system symmetry

4-fold rotation

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Hexagonal crystal system symmetry

3- or 6-fold rotation

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a, b, and y in oblique 2D unit cells

all unrestricted

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a, b, and y in rectangular 2D unit cells

a and b unrestricted
y = 90 degrees

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a, b, and y in square 2D unit cells

a = b
y = 90 degrees

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a, b, and y in hexagonal 2D unit cells

a = b
y = 120 degrees

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Filling the oblique unit lattice with a molecule creates an

oblique unit cell

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Primitive unit cell

Lattice points only at the vertices

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Centered unit cell

Lattice points at places other than vertices

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Unit lattice + motif =

unit cell

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Protein crystals are so fragile and sensitive to environmental changes because

only a few contacts exist within the crystal

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Only ______ and ______ unit cells allow an entirely arbitrary choice of origin

primitive p1 and P1

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Any crystallographic symmetry operation must generate

an identical copy of the motif

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Translational restrictions limit all crystallographic rotation operations to

2-, 3-, 4-, and 6-fold rotations

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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

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?

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

High hkl provide ______ detail about the sampled structure