2' Principles of Xray crystallography

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2. Principles of X-ray crystallography
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The Seven Crystal Systems (1)
Define unit cell shapes. Must fill 3D space
this imposes constraints on allowed unit cell
symmetry
For example it is not possible to fill 2D space
with pentagons, so it is not possible to have
five-fold crystallographic symmetry.
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Importance of Symmetry

• Model a crystal structure with atomic
• positions.
• Making maximum use of symmetry
• minimizes the number of parameters to be
• determined.
• Crystal system defines the maximum
• amount of symmetry that can expressed by a
  crystal structure.

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The Concept in Molecules
This molecule has D4h symmetry. Therefore the
coordinates of all F-atoms can be derived
from those of just one.
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Ammonium oxalate hydrate

• Cell contains 38 atoms, each with 3
• coordinates (114 positional parameters to
• determine)
• If symmetry is taken into account this
• reduces to 28.

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The Seven Crystal Systems (2)
only seven possible shapes of three dimensional
unit cells crystal systems
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Lattice centering

• Primitive P
• Face-centered F, A, B, C
• Body centered I

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Primitive vs. centered
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Centered Lattices

• In order to make maximum use of symmetry it is
  sometimes necessary to double the area of a 2D
  unit cell (or 2x, 3x, 4x for a 3D unit cell).

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Not all centered lattices are unique. Consider a
centered square lattice
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Not all centered lattices are unique. Consider a
centered square lattice
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14 Bravais Lattices
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Crystal Planes and Miller Indexes
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The right-handed rule

• Thumb is x, forefinger is y and middle finger is
  z.
• Right handed screw motions are x into y makes
  a positive motion into z. It follows that y
  into z is a x screw and z into x is a z
  screw.
• Whenever in doubt, pull your hand out of your
  pocket to set up coordinate systems. Left handed
  coordinate systems lead to negative cell volumes!

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Crystal Planes and Miller Indexes Planes and
directions
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Crystal Planes and Miller Indexes Equivalent
planes and directions
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Animations

• Miller Indexes
• Equivalent planes

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X-ray powder diffraction

• Identification of compounds
• Crystallinity
• Phases
• Wide uses in chemistry, minerals, material
  sciences

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Reciprocal lattice (1)

• P. P. Ewald in 1913
• The points of a reciprocal lattice represent the
  planes of the direct (i.e. real) lattice that it
  is formed from.
• The direct lattice determines (through defined
  relationships) the reciprocal lattice vectors,
  the lattice point spacing and the associated
  reciprocal directions.

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Reciprocal lattice (2)
The reciprocal lattice has reciprocal vectors a
and b, separated by the angle g. a will be
perpendicular to the (100) planes, and equal in
magnitude to the inverse of d100. Similarly, b
will be perpendicular to the (010) planes and
equal in magnitude to the inverse of d010. Hence
g and g will sum to 180º.
Consider the two dimensional direct lattice shown
below. It is defined by the real vectors a and b,
and the angle g. The spacings of the (100) and
(010) planes (i.e. d100 and d010) are shown.
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Reciprocal lattice (3)

• Due to the linear relationship between planes
  (for example, d200 (1/2)d100), a periodic
  lattice is generated. In general, the periodicity
  in the reciprocal lattice is given by
• In vector form, the general reciprocal lattice
  vector for the (hkl) plane is given by
• where nhkl is the unit vector normal to the (hkl)
  planes.

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Array of points making up the reciprocal lattice
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For non-primitive lattices, such as a C-centred
lattice, systematic absences can occur in the
reciprocal lattice and in the diffraction
patterns. This is due to the construction of the
lattices.
The reciprocal lattice is now constructed using
the different lattice vectors and interplanar
spacings. When it is labelled with respect to the
new reciprocal lattice vectors, the dashed spots
are "absent".
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Systematic Absence

• These absences help distinguish different crystal
  lattice types from the diffraction patterns,
  since each type has a characteristic pattern of
  absences. In this example, points with ( h k )
  as an odd integer are absent, due to the
  definition of the unit cell.

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The Ewald Sphere

• Consider a circle of radius r, with points X and
  Y lying on the circumference.

If the angle XAY is defined as q, then the angle
XOY will be 2q by geometry. Also, sin q
XY/2r If this geometry is constructed in
reciprocal space, then it has some important
implications.
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• The radius can be set to 1/l, where l is the
  wavelength of the X-ray beam.
• If Y is the 000 reciprocal lattice point, and X
  is a general point hkl, then the distance XY is
  1/dhkl
• Hence
• i.e.
• l 2 dhkl sin q
• This is Bragg's Law. Effectively, the application
  of this circle to the reciprocal lattice defines
  the points which satisfy Braggs Law (X on the
  diagram). Therefore the (hkl) planes
  corresponding to these reciprocal points will
  diffract X-rays of wavelength l at the angle q.

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• Crystal lattices are three-dimensional, and hence
  so are their reciprocal lattices. The necessary
  circle is now a sphere. This is known as the
  Ewald sphere.

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• The Ewald sphere construction shares the
  properties of Bragg's law.

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• Diffraction occurs when a reciprocal lattice
  point intersects the Ewald sphere.

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• The "limiting sphere sphere represents the limit
  of resolution of your crystal. So for a crystal
  diffracting to 2 Å this sphere would have a
  radius of 1/2. All reciprocal lattice points
  within this sphere can in principle be made to
  diffract by letting them intersect the Ewald
  sphere.

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Resolution and disorder
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View a step by step construction of the Ewald
sphere.

• http//www.msm.cam.ac.uk/doitpoms/tlplib/DD2/ewald
  .php

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Ewalds sphere construction in 3D

• http//www.matter.org.uk/diffraction/geometry/ewal
  d_sphere_construction_3D.htm

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Limiting sphere the complete data set
If one rotates the Ewald sphere completely about
the (000) reciprocal lattice point in all three
dimensions, the larger sphere (of radius 2/?)
contains all of the reflections that it is
possible to collect using that wavelength of
X-rays. This construction is known as the
Limiting sphere and it defines the complete
data set. Any reciprocal lattice points outside
of this sphere can not be observed.
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• Note that the shorter the wavelength of the
  X-radiation, the larger the Ewald sphere and the
  more reflections may be seen (in theory).
• The limiting sphere will hold roughly (4/3pr3/
  V) lattice points. Since r 2/?, this equates
  to around (33.5/ V?3) or (33.5 V/?3)
  reflections.
• For an orthorhombic cell with a volume of
  1600Å3, this means CuKacan give around 14,700
  reflections while MoKawould give 152,000.

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The Sphere of Reflection and Braggs Law
Copper radiation used for macromolecular
structures of large unit cell and organic
molecules. Diffracted beams are more separated on
a detector if a longer wavelength is used.
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A reciprocal lattice plot
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h
Different sets of planes in the crystal give rise
to different diffraction spots.
Circles of constant theta (resolution)
k
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x
O
y
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x
O
(100) planes
y
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h
k
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x
O
(200) planes
y
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X-axis cut At 1
Y-axis cut at 1/2
Z-axis not cut at all Cut at ?.
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(1,2,0) planes
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h
k
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h
k
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Direct Lattice vs. Reciprocal Lattice

• Recall that X-rays reflect from electrons.
• The direct lattice representation shown earlier
  is filled with atoms and molecules which diffract
  X-rays.
• The reciprocal lattice adopts an inverse motif
  where the axes are measured in Å-1. The volume
  between the reciprocal lattice vertices is void.
  The relative intensity is marked at integer
  indices, e.g., (123).
• Braggs Law, 2dsin? n?, can be rearranged to
  test for the appearance of reflections. The
  general form for Braggs Law in reciprocal space
  is sin? n?/ 2d. This means that diffracted beam
  is redirected by a known value for a certain
  d-spacing and X-ray wavelength.

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Orthorhombic Direct and Reciprocal Cell
Relationships
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Monoclinic Direct and Reciprocal Cell
Relationships
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Triclinic Direct and Reciprocal Cell
Relationships
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Questions

• http//www.msm.cam.ac.uk/doitpoms/tlplib/DD2/quest
  ions.php