Scattering of EM Radiation Light

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Scattering of EM Radiation (Light) Charged
particles are accelerated by electric
fields. Charge particles are the source of
electric fields Acceleration of a charged
particle perturbs the electric field.
Accelerating electrons radiate photons! When an
electron interacts with a electromagnetic wave
it oscillates at the same frequency of the
wave. Generates electromagnetic radiation with
the same frequency 180o out of phase. - This
is called scattering.
Oscillation direction of the electron
?
X-ray beam
Electron
Electric vector of The incident beam
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Some common terms use with crystals
Molecule the protein (or nucleic acid) that is
crystallized. Asymmetric unit the smallest unit
that can be rotated and translated to generate
one unit cell using only the symmetry operators
allowed by the crystallographic symmetry. Unit
cell the smallest repeating unit that can
generate the crystal with only translation
operations. Crystal a regular repeat of
molecules, usually with some sort of internal
rotational symmetry. Protein crystals are
usually about 40-60 solvent by weight and are
thus fragile and sensitive to drying out.
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The Basics Unit Cell, Miller Indices Lattices
Unit cell Unit cell is the smallest unit of a
crystal, which, if repeated, could generate the
whole crystal.
A crystals unit cell dimensions are defined by
six numbers, the lengths of the 3 axes, a, b, and
c, and the three interaxial angles, ?, ? and
?. To generate a crystal lattice, simply repeat
the unit cell in all directions. By definition,
items in any unit cell are related to those in
any other unit cell simply by translational
symmetry (moving along x, y, and/or z).
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Local vs. Global Symmetry
Crystallographicsymmetry Applies to both
Molecules lattice Reflections related to one
another (formally) Need to collect only a
subset of data Recognition constrains possible
structures Enough to solve very simple
structures Failure to recognize -gt elevated
degrees of freedom -gt incorrect structures
sometimes Diffraction has only just enough
information to solve unique part of structure
Non-crystallographic symmetry Applies to
molecules or contents of unit cell, but not
lattice. No simple relationships between
diffraction intensities Symmetrical molecules
are the same More favorable ratio of data to
model parameters Can be used to develop
constraints on phases ...
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The Basics Unit Cell, Miller Indices Lattices
A crystal lattice is a 3-D stack of unit cells.
Crystal lattice is an imaginative grid system in
three dimensions in which every point (or node)
has an environment that is identical to that of
any other point or node. Any position within the
crystal can be described by r (u x) a (v
y) b (w z) c where u, v w are integers
0 lt x, y, z lt 1. x, y z are called
fractional coordinates describe a position
within the unit cell.
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A two dimensional lattice representation
Consider a two-dimensional plane. A number of
different planes with different orientations may
be drawn, with each plane containing some of the
lattice points. Each line or plane is a
representative number of a parallel set of
equally spaced lines or planes. Each lattice
point must lie on one of the lines or
planes. These lattice lines and planes are
labelled using Miller Indices.
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A two dimensional lattice representation
Some basic rules All planes in a set are
identical The planes are imaginary The
perpendicular distance between pairs of adjacent
planes is the d-spacing from Braggs lab (n?
2d sin??).
For a three- dimensional unit cell, three indices
are required and are conventionally designated
h, k and l. The Miller indices for a particular
family of planes are usually written (h,k,l),
where h, k and l are positive or negative
integers or zero.
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Miller Indices and Reciprocal Space
To deduce the Miller indices, consider how the
lattice is intersected by the set of (imaginary)
planes. Then, take the reciprocal on
these numbers. Recall that the unit cell shown
here as defined by the lengths a, b, c and the
angles ?, ? and ?.
In this example, the plane shown above (in red)
cuts the a length (along the x-axis) at
1/2. The same plane cuts the b length (along y)
just once and the c length (along z) at
1/2. Thus, the Miller indices for this set of
planes would be (2 1 2).
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Miller Indices and Reciprocal Space
In this example, the set of planes cut the a
length into 1/2, the b length just once and the
c length into 1/3.
Thus, the Miller indices for this set of planes
would be (2 1 3). The Miller indices of any
particular family of planes are given by the
reciprocals of the fractional intercepts along
each of the cell directions. It is possible to
have the planes cut the unit cell into
non-integral lengths (such as 3/2). In this
instance, multiply the reciprocal lengths until
you get integers (example (4 3/2 2) would be (8
3 4).
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Miller Indices and Reciprocal Space
Miller indices define the orientation of the
plane within the unit cell If a set of planes is
perpendicular to any of the axes, it would cut
that axes at ?, hence the Miller index along that
direction is 1/ ? 0. If a plane to be
indexed has an intercept along the negative
portion of a coordinate axis, a minus sign is
placed over the corresponding index. The Miller
Index defines a set of planes parallel to one
another (remember the unit cell is a subset of
the infinite crystal (002) planes are parallel
to (001) planes, and so on.
Plane perpendicular to y cuts at ?, 1, ? ? (0 1
0) plane
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Miller Indices and d Spacing
In the diagram, look at Plane B (with Miller
indices (1 2). To calculate the interplanar
spacing between two such planes, first consider
the right triangle with sides 2a and b. Use this
to calculate the angle ? tan (?)
b/2a Now, use the derived value of ? to
calculate d sin (?) d /2a All this can be
concisely be derived using the d-spacing formula.
2a
b
For orthogonal crystal systems (i.e. ???90?)
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Diffraction - an optical grating
Path difference XY between diffracted beams 1 and
2 sin? XY/a ? XY a sin ?
For 1 and 2 to be in phase and give constructive
interference, XY ?, 2?, 3?, 4?..n? so a
sin ? n? where n is the order of
diffraction
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Diffraction - an optical grating
Consequences maximum value of ? for
diffraction sin? 1 ? a ? Realistically, sin
?lt1 ? a gt ? So separation must be same order as,
but greater than, wavelength of light. Thus for
diffraction from crystals Interatomic distances
0.1 - 2 Å so ? 0.1 - 2 Å X-rays, electrons,
neutrons suitable
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Constructive Interference
Beam 2 lags beam 1 by XYZ 2d sin ? so 2d sin
? n? Braggs Law
If X-ray using a home source are diffracted to
Bragg spacings of 1.2 Å, calculate the Bragg
angle, ?, for constructive interference.
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Constructive Interference
If X-ray using a home source are diffracted to
Bragg spacings of 1.2 Å, calculate the Bragg
angle, ?, for constructive interference.
? 1.54 x 10-10 m, d 1.2 x 10-10 m, ??
n1 ? 39.9 n2 X (n?/2d)gt1
2d sin ? n?
We normally set n1 and adjust Miller indices, to
give 2dhkl sin ? ?
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Conclusion about Miller Indices
We can imagine planes within a crystal. Each set
of planes is uniquely identified by its Miller
index (h k l) We can calculate the separation,
d, for each set of planes (h k l) Crystals
diffract radiation of a similar order of
wavelength to the interatomic spacings. We model
this diffraction by considering the reflection
of radiation from planes - Braggs Law Each
diffraction spot can be regarded as a X-ray beam
reflected from a lattice plane, and therefore
has a unique Miller index.
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Crystallographic Symmetry a Primer

• Symmetry operations are geometrically defined
  ways of exchanging
• equivalent parts of a molecule. There are 4
  kinds that are typically used
• Simple rotation about an axis passing
• through the molecule by an angle 2?/n.
• This operation is called a proper rotation.
• If it is repeated n times then the molecule
• returns to its original orientation.
• 2. Reflection of all atoms through a plane which
  passes through the
• molecule. This operation is called reflection.
• 3. Reflection of all atoms through a point in the
  molecule. This operation
• is called inversion.
• 4. The combination of rotating the
• molecule about an axis passing
• through it by 2 ? /n and reflecting all
• atoms through a plane perpendicular

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Crystallographic Symmetry a Primer
The use of these operations are symmetry
operations if, and only if, the appearance of
the molecule is exactly the same after one of
them is carried out as it was before. Symmetry
a state in which parts on opposite sides of a
plane, line, or point display arrangements that
are related to one another via a symmetry
operation such as translation, rotation,
reflection or inversion. Application of the
symmetry operators leaves the entire crystal
unchanged.
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Crystallographic symmetry Rotations
Symmetry Elements Rotational symmetry
Twofold rotation (360o/2) written 2 or Threefold
rotation (360o/3) written 3 or Fourfold rotation
(360o/4) written 4 or Six-fold rotation (360o/6)
written 6 or
Why dont we deal with 5-fold or 7-fold
symmetry? These symmetry elements never occur in
Nature. It is impossible to pack something with
5-fold or gt 6-fold symmetry
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Crystallographic symmetry Rotations
Which rotations are consistent with a
lattice? Suppose f 360o/n. Any valid symmetry
operation must result in landing on a lattice
point. In order to satisfy this definition
2a cos f u a since it must be possible
to get from one point to the next by the lattice
translation symmetry. Only possible solutions
are u -2, -1, 0, 1 or 2 only.
2 a cos f
f
Rotates to here
a
f
a
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Crystallographic symmetry Rotations
In order to satisfy this definition 2a cos
f u a since it must be possible to get
from one point to the next by the lattice
translation symmetry. Only possible solutions
are u -2, -1, 0, 1 or 2 only.

• Only 1, 2, 3, 4 6 rotations consistant with
  translational symmetry.
• Eg. 5 or 10 are never seen as crystal rotational
  symmetries!

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• Rotation matrix.
• r2 Rn r1.
• Eg. For rotational symmetry 2 about the y axis
• Hence (x, y, z) ? (-x, y, -z)

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• Mirror planes
• The mirror plane normal is parallel with an axis
  of the lattice.
• If the mirror plane is parallel to z
• r2 Mz r1.
• where
• Hence (x, y, z) ? (x, y, -z).

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Crystallographic symmetry improper rotations
Symmetry Elements Screw axes (rotation
translation)
Rotation about the axis of symmetry by 360?/n,
followed by a translation parallel to the axis by
r/n of the unit cell length in that direction. (r
lt n)
A 31 screw axis would involve a rotation of 360/3
(120 degrees) about the screw axis followed by a
translation of 1/3 the unit cell in that
direction
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Crystallographic symmetry screw axes
Compare 31 symmetry with a 32 symmetry. Each one
involves a rotation of 120 degrees about the
screw axis, but the 31 involves a 1/3 translation
along the screw axis while the 32 involves a 2/3
translation.

• 21 means translate 1/2 of the unit cell rotate
  180o
• 31 means translate 1/3 of the unit cell rotate
  120o.
• 42 means translate 2/4 of the unit cell rotate
  90o.
• 41 means translate 1/4 of the unit cell rotate
  90o.

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Seven Crystal Systems
Crystals are grouped into seven crystal systems,
according to characteristic symmetry of their
unit cell. The characteristic symmetry of a
crystal is a combination of one or more
rotations and inversions.
hexagonal
orthorhombic
monoclinic
trigonal
cubic
tetragonal
triclinic
Crystal System External Minimum Symmetry Unit
Cell Properties Triclinic None a, b, c, ?,
?, ? Monoclinic One 2-fold axis, to b (b
unique) a, b, c, 90, ?, 90 Orthorhombic Three
perpendicular 2-folds a, b, c, 90, 90,
90 Tetragonal One 4-fold axis, parallel c a, a,
c, 90, 90, 90 Trigonal One 3-fold axis a, a,
c, 90, 90, 120 Hexagonal One 6-fold axis a, a,
c, 90, 90, 120 Cubic Four 3-folds along space
diagonal a, a, ,a, 90, 90, 90
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• The Bravais Lattices
• Four clases of centering types
• - Primitive unit cell (P).
• - Plane centred unit cell (A, B or C).
• - Body centred unit cell (I).
• - Face centred unit cell (F).
• In combination with previous 7 primitive cells
  leads to 14 Bravais lattices.

F
P
A
I
A Bravais lattice is an infinite array of
discrete points with identical environments.
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The Bravais Lattices
Why bother with non-primitive cells?

• The blue unit-cell is a primitive unit cells
  with ½ the volume of the B unit cell.
• - However some symmetry of the total unit cell is
  lost in making the choice of the blue cell.
• - a ? b ? g in this choice but a b g 90o in
  the other.

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Point group symmetry

• Inorganic crystals usually have perfect shape
  which reflects their internal symmetry
• Point groups are originally used to describe the
  symmetry of crystal.
• Point group symmetry does not consider
  translation.
• Included symmetry elements are rotation, mirror
  plane, center of symmetry, rotary inversion.

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There are a total of 32 point groups
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Space groups
The combination of all available symmetry
operations (32 point groups) within the all
available lattices (14 Bravais lattices) lead to
230 Space Groups. These space group describe the
only ways in which identical objects can be
arranged in an infinite lattice. The
International Tables list those by symbol and
number (symmetry operators, origins, reflection
conditions, and space group projections).
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Notation Space groups are first labelled
according to lattice ie. P, A (B or C), F or
I. Next labelled relative to the highest
symmetry axes. - P6 means a primitive lattice
with 1 6-fold axis. - P222 means a primitive
lattice with three two-fold axes. - F means a
face-centred lattice with no additional symmetry.
A subscript is then used to denote screw
axes. - P63 means a six-fold screw axis. -
P212121 means three two-fold screw axes.
Point group 2 (2-fold symmetry) Bravais lattice
P (primitive) Screw axis along 2-fold
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Space group determination

• Symmetry in diffraction pattern
• If the crystal has n-fold symmetry, the
  diffraction pattern will also have n-fold
  symmetry.
• Systematic absences
• If a crystal has a 2-fold screw axis, the
  symmetry of the diffraction pattern will be (x,
  y, z) (-x, y1/2, -z). Since the symmetry
  position along the y-axis also involves a
  translational component, the diffraction pattern
  will have systematic absences. We will prove
  this later on.
• Space groups with mirror planes and inversion
  centers do not apply to protein crystals, leaving
  only 65 possible space groups.

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• Equivalent positions
• Suppose (x, y, z) is a normal fractional
  coordinate.
• For each symmetry operation within the
  space-group there exists another position (x,
  y, z) within the unit cell which is rellated to
  (x, y, z) by symmetry.
• (x, y, z) is called an equivalent position.

• eg. 2-fold rotation about a takes (x, y, z) out
  of the unit cell.
• A translation by b c returns it into the unit
  cell.
• This is (x, y, z) is an equivalent position
  to (x, y, z)

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Space group determination

• Symmetry in diffraction pattern
• If the crystal has n-fold symmetry, the
  diffraction pattern will also have n-fold
  symmetry.
• Systematic absences
• If a crystal has a 2-fold screw axis, the
  symmetry of the diffraction pattern will be (x,
  y, z) (-x, y1/2, -z). Since the symmetry
  position along the y-axis also involves a
  translational component, the diffraction pattern
  will have systematic absences. We will prove
  this later on.
• Space groups with mirror planes and inversion
  centers do not apply to protein crystals, leaving
  only 65 possible space groups.

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