Xrd

INTRODUCTION TO XRD

Unit cell: the building block of crystalline solids z

The parameters that define a unit cell are: a, b, c = unit cell dimensions along x, y, z respectively α, β, γ = angles between b,c (α); a,c (β); a,b (γ)

x

y

Shapes of unit cells All the possible shapes of a unit cell are defined by 7 crystal systems, which are based on the relationship among a,b,c and α, β, γ,

a = b = c ; α = β = γ = 90°

Cubic system

a = b ≠ c ; α = β = γ = 90°

Tetragonal system

a ≠ b ≠ c ; α = β = γ = 90°

Orthorhombic system

a = b = c ; α = β = γ ≠ 90°

Rhombohedral system

a = b ≠ c ; α = β = 90°; γ = 120°

Hexagonal system

a ≠ b ≠ c ; α = γ = 90° ≠ β

Monoclinic system

a ≠ b ≠ c ; α ≠ β ≠ γ ≠ 90°

Triclinic system

Types of Unit Cells

Primitive lattice: the unit cell has a lattice point at each corner only (P)

Body centred lattice: the unit cell has a lattice point at each corner and one in the centre (I)

Face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of one pair of opposite faces (A), (B), (C)

All-face-centred lattice: the unit cell has a lattice point at each corner and one in the centre of each face (F)

CHARACTERIZATION OF THE STRUCTURE OF SOLIDS Three main techniques: X-ray diffraction Single crystal Powder

Electron diffraction

Neutron diffraction

Principles of x-ray diffraction X-rays are passed through a crystalline material and the patterns produced give information of size and shape of the unit cell X-rays passing through a crystal will be bent at various angles: this process is called diffraction X-rays interact with electrons in matter, i.e. are scattered by the electron clouds of atoms

WHAT IS DIFFRACTION? Diffraction – the spreading out of waves as they encounter a barrier.

What is a Diffraction pattern? -

an interference pattern that results from the superposition of waves.

-

Mathematically, this process can be described by Fourier transform, if the diffraction is kinematic (electron or X-ray has been scatted only once inside the object). Laser diffraction pattern of a thin grating films, where the size of holes is closed to the wavelength of the laser (Ruby red light 594 um) .

Fourier transform of regular lattices:

Real space

Reciprocal space

INTERACTION BETWEEN XRAY AND MATTER d incoherent scattering

λCo (Compton-Scattering) wavelength λPr intensity Io

coherent scattering

λPr(Bragg´s-scattering) absorbtion Beer´s law I = I0*e-µd fluorescense

λ> λPr

photoelectrons

The angles at which x-rays are diffracted depends on the distance between adjacent layers of atoms or ions. X-rays that hit adjacent layers can add their energies constructively when they are “in phase”. This produces dark dots on a detector plate

Scattering of x-rays by crystallographic planes We need to consider how x-rays are diffracted by parallel crystallographic planes

incident x-rays

lattice planes

λ

diffracted x-rays

d

atoms on lattice planes X-rays diffracted in phase will give a signal. “In phase” means that the peak of one wave matches the peak of the following wave

The angle of incidence of the x-rays is is θ The angle at which the xrays are diffracted is equal to the angle of incidence, θ

θ C D

θ θ

F E

d

The angle of diffraction is the sum of these two angles, 2θ

The two x-ray beams travel at different distances. This difference is related to the distance between parallel planes We connect the two beams with perpendicular lines (CD and CF) and obtain two equivalent right triangles. CE = d (interplanar distance)

DE sin θ = d

d sin θ = DE

DE = EF

d sin θ = EF

2d sin θ = EF + DE = difference in path length

Reflection (signal) only occurs when conditions for constructive interference between the beams are met These conditions are met when the difference in path length equals an integral number of wavelengths, n. The final equation is the BRAGG’S LAW

nλ = 2d sin θ Data are collected by using x-rays of a known wavelength. The position of the sample is varied so that the angle of diffraction changes When the angle is correct for diffraction a signal is recorded

Intensity (a.u.)

With modern x-ray diffractometers the signals are converted into peaks (301) (310)

(200)

(600) (411)

(110) (400)

2θ degrees

(611) (321) (002)

TEST NaCl is used to test diffractometers. The distance between a set of planes in NaCl is 564.02 pm. Using an x-ray source of 75 pm, at what diffraction angle (2θ) should peaks be recorded for the first order of diffraction (n = 1) ? Hint: To calculate the angle θ from sin θ, the sin-1 function on the calculator must be used

nλ = 2d sin θ 1 × 75 pm = 2 × 564.02 pm × sin θ 75 pm sin θ = = 0.066 2 × 564.02 pm

θ = 3.81 ; 2θ = 7.62 

Lattice Planes and Miller Indices Atoms or ions in lattices can be thought of as being connected by lattice planes. Each plane is a representative member of a parallel set of equally spaced planes. A family of crystallographic planes is always uniquely defined by three indices, h, k, l, (Miller indices) usually written (h, k, l)

h=

The Miller indices are defined by

1 1 1 ,k = ,l = X Y Z

X, Y, Z are the intersections of one plane with on a, b, c respectively

x

( 0kl ) ( h0l ) ( hk 0 )

family of lattice planes parallel to

y z

Note ­ plane // to axis, intercept = ∞ and  1/∞ = 0

How to Determine Miller Indices

EXAMPLES OF CRYSTALLOGRAPHIC PLANES c

c

(111)

b

b a

a c

(212)

0.5 b a

(100)

Inter-Planar Spacing, dhkl, and Miller Indices The inter-planar spacing (dhkl) between crystallographic planes belonging to the same family (h,k,l) is denoted (dhkl) Distances between planes defined by the same set of Miller indices are unique for each material

dhkl

2D d'h’k’l’

Inter-planar spacings can be measured by x-ray diffraction (Bragg’s Law)

The lattice parameters a, b, c of a unit cell can then be calculated The relationship between d and the lattice parameters can be determined geometrically and depends on the crystal system Crystal system

dhkl, lattice parameters and Miller indices

Cubic

1 h2 + k 2 + l 2 = 2 d a2

Tetragonal

1 h2 + k 2 l 2 = + 2 2 2 d a c

Orthorhombic

1 h2 k 2 l 2 = 2 + 2 + 2 2 d a b c

The expressions for the remaining crystal systems are more complex

THE POWDER TECHNIQUE An x-ray beam diffracted from a lattice plane can be detected when the x-ray source, the sample and the detector are correctly oriented to give Bragg diffraction A powder or polycrystalline sample contains an enormous number of small crystallites, which will adopt all possible orientations randomly Thus for each possible diffraction angle there are crystals oriented correctly for Bragg diffraction

Each set of planes in a crystal will give rise to a cone of diffraction

Each cone consists of a set of closely spaced dots each one of which represents a diffraction from a single crystallite

FORMATION OF A POWDER PATTERN Single set of planes

Powder sample

Experimental Methods To obtain x-ray diffraction data, the diffraction angles of the various cones, 2θ, must be determined The main techniques are: Debye-Scherrer camera (photographic film) or powder diffractometer Debye Scherrer Camera

Powder Diffractometer

The detector records the angles at which the families of lattice planes scatter (diffract) the x-ray beams and the intensities of the diffracted x-ray beams The detector is scanned around the sample along a circle, in order to collect all the diffracted x-ray beams The angular positions (2θ) and intensities of the diffracted peaks of radiation (reflections or peaks) produce a two dimensional pattern Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl) This pattern is characteristic of the material analysed (fingerprint)

Intensity

(301) (310)

(200)

(600) (411)

(110) (400)

2θ degrees

(611) (321) (002)

APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA Information is gained from: crystal class lattice type

Number and positions (2θ) of peaks

cell parameters Intensity of peaks

types of atoms position of atoms

Identification of unknown phases Determination of phase purity Determination and refinement of lattice parameters Determination of crystallite size Structure refinement Investigation of phase changes

Identification of compounds The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS) ‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database This technique can be used in a variety of ways

PDF - Powder Diffraction File A collection of patterns of inorganic and organic compounds Data are added annually (2008 database contains 211,107 entries)

Example of Search-Match Routine

Outcomes of solid state reactions

2SrCO3 + CuO

Product: SrCuO2? Pattern for SrCuO2from database

Product: Sr2CuO3? Pattern for Sr2CuO3from database

?

SrCuO2 Sr2CuO3

Phase purity When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern) Sr2CuO2F2+δ

Sr2CuO2F2+δ + impurity

*

Effect of defects 8 mol % Y2O3 in ZrO2 (cubic)

3 mol % Y2O3 in ZrO2 (tetragonal)

ZrO2 (monoclinic)

http://www.talmaterials.com/technew.htm

Determination of crystal class and lattice parameters X-ray powder diffraction provides information on the crystal class of the unit cell (cubic, tetragonal, etc) and its parameters (a, b, c) for unknown compounds

1

Crystal class

2

Indexing

3

Determination of lattice parameters

Cubic system

comparison of the diffractogram of the unknown compound with diffractograms of known compounds (PDF database, calculated patterns)

Assigning Miller indices to peaks

Bragg equation and lattice parameters

2 λ sin 2 θ = 2 ( h 2 + k 2 + l 2 ) 4a

PROBLEM NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices (λ = 1.54056 Å). ?

?

Selected data from the NaCl diffractogram 2θ (°)

h,k,l

27.47

111

31.82

?

45.62

?

56.47

222

2 λ sin 2 θ = 2 ( h 2 + k 2 + l 2 ) 4a

a (Å)

Use at least two reflections and then average the results

λ2 ( h 2 + k 2 + l 2 ) 1.5412 ×12 a= = = 5.638 Å 2 56.473 4 sin θ 4 × sin 2 2

Miller Indices

λ2 2 sin θ = 2 ( h + k 2 + l 2 ) 4a 2

A

(

sin 2 θ = A h 2 + k 2 + l 2

)

(222)

λ2 1.54056 2 A= 2 = = 0.01867 2 4a 4 × ( 5.638) sin 2 θ = h2 + k 2 + l 2 A

(

2θ = 31.82



sin 2 31.82

2 = 4.026 ≈ 4

(h

2 = 8.052 ≈ 8

(h

0.01867

2θ = 45.62



sin 2 45.62 0.01867

) 2

2

)

+ k 2 + l 2 = 4 ∴ ( 200 )

)

+ k 2 + l 2 = 8 ∴ ( 220 )

Systematic Absences For body centred (I) and all-face centred (F) lattices restriction on reflections from certain families of planes, (h,k,l) occur. This means that certain reflections do not appear in diffractograms due to ‘out-of-phase” diffraction This phenomenon is known as systematic absences and it is used to identify the type of unit cell of the analysed solid. There are no systematic absences for primitive lattices (P)

Conditions for reflection

F (h

2

)(

)(

i.e indices are all odd or all even

I P

(h

)

+ k 2 , h 2 + l 2 , k 2 + l 2 = 2n (even number) 2

+ k + l ) = 2n 2

2

No conditions

Considering systematic absences, assign the following sets of Miller indices to either the correct lattice(s).

Lattice Type Miller Indices

P

I

F

100

Y

N

N

110

Y

Y

N

111

Y

N

Y

200

Y

Y

Y

210

Y

N

N

211

Y

Y

N

220

Y

Y

Y

310

Y

Y

N

311

y

N

Y

Autoindexing Generally indexing is achieved using a computer program. This process is called ‘autoindexing’ Input:

•Peak positions (ideally 20-30 peaks) •Wavelength (usually λ=1.54056 Å) •The uncertainty in the peak positions •Maximum allowable unit cell volume

Problems:

•Impurities •Sample displacement •Peak overlap

Derivation of

λ2 2 sin θ = 2 ( h + k 2 + l 2 ) 4a 2

1 h2 + k 2 + l 2 = 2 d a2 2

a d = 2 ; 2 2 h + k +l 2

λ = 2d sin θ 1 d =a 2 h + k2 + l2

1 λ = 2a 2 sin θ 2 2 h + k +l λ sin θ = h2 + k 2 + l 2 2a 2 λ sin2θ = 2 h 2 + k 2 + l 2 4a