1

2. Principles of X-ray crystallography

2

The Seven Crystal Systems (1)

• Define unit cell shapes.• Must fill 3D space – this imposesconstraints on allowed unit cell symmetry:

For example: it is not possibleto fill 2D space withpentagons, so it is notpossible to have five-foldcrystallographic symmetry.

3

Importance of Symmetry

• Model a crystal structure with atomicpositions.

• Making maximum use of symmetryminimizes the number of parameters to bedetermined.

• Crystal system defines the maximumamount of symmetry that can expressed by a crystal structure.

4

The Concept in Molecules

This molecule has D4h symmetry.Therefore the coordinates ofall F-atoms can be derived fromthose of just one.

5

6

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.

7

The Seven Crystal Systems (2)

only seven possible shapes of three dimensional unit cells; crystal systems

8

Lattice centering

• Primitive; P

• Face-centered; F, A, B, C

• Body centered; I

9

Primitive vs. centered

10

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).

11

Not all centered lattices are unique. Consider a centered square lattice:

12

Not all centered lattices are unique. Consider a centered square lattice:

13

14 Bravais Lattices

14

Crystal Planes and Miller Indexes

15

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!

16

Crystal Planes and Miller Indexes: Planes and directions

17

Crystal Planes and Miller Indexes: Equivalent planes and directions

18

Animations

• Miller Indexes

• Equivalent planes

19

X-ray powder diffraction

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

20

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.

21

Reciprocal lattice (2)

Consider the two dimensional direct lattice shown below. It is defined by the real vectors a and b, and the angle . The spacings of the (100) and (010) planes (i.e. d

100 and d010) are shown.

The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle *. 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 and * will sum to 180º.

22

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.

23

Array of points making up the reciprocal lattice

24

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".

25

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.

26

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 , then the angle XOY will be 2by geometry. Also,sin = XY/2r

If this geometry is constructed in reciprocal space, then it has some important implications.

27

• The radius can be set to 1/, where 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. = 2 dhkl sin

• 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 at the angle .

28

• 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.

29

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

30

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

31

• 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.

32

Resolution and disorder

33

View a step by step construction of the Ewald sphere.

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

34

Ewald’s sphere construction in 3D

• http://www.matter.org.uk/diffraction/geometry/ewald_sphere_construction_3D.htm

35

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/λ) 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.

36

• 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 (4/3πr3/ V*) lattice points. Since r = 2/λ, this equates to around (33.5/ V*λ3) or (33.5 V/λ3) reflections.

• For an orthorhombic cell with a volume of 1600Å3, this means CuKαcan give around 14,700 reflections while MoKαwould give 152,000.

37

The Sphere of Reflection and Bragg’s Law

Copper radiation used for Copper radiation used for macromolecular structuremacromolecular structures of large unit cell and orgs of large unit cell and organic molecules. Diffracted anic molecules. Diffracted beams are more separatebeams are more separated on a detector if a longer d on a detector if a longer wavelength is used.wavelength is used.

38

A reciprocal lattice plot

39

h

k

Circles of constanttheta (resolution)

Different sets of planesin the crystal give rise todifferent diffraction spots.

40

y

xO

41

y

xO

(100) planes

42

h

k

43

y

xO

(200) planes

44

y

xO

45

y

xO

X-axis cutAt +1Y-axis cut

at 1/2

Z-axis not cut at all = Cut at .

46

y

xO

(1,2,0) planes

47

h

k

48

h

k

49

Direct Lattice vs. Reciprocal Lattice

• Recall that X-rays reflect from electrons.• The direct lattice representation shown earlier is filled wit

h 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).

• Bragg’s Law, 2dsinθ =nλ, can be rearranged to test for the appearance of reflections. The general form for Bragg’s Law in reciprocal space is sinθ =nλ/ 2d. This means that diffracted beam is redirected by a known value for a certain d-spacing and X-ray wavelength.

50

Orthorhombic Direct and Reciprocal Cell Relationships

51

Monoclinic Direct and Reciprocal Cell Relationships

52

Triclinic Direct and Reciprocal Cell Relationships

53

Questions

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