mle2101 lab report

MLE2101 Lab Report:

X-Ray Crystallography

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MLE2101

Introduction to Structure of Materials


Lai Yin Kee (A0083696M)

Lee Shao Cheng Alex (A0086420L)

Lee Yong Kiat (A0086953N)

Lim Yee Ching (A0085222M)

Lim Ze Ming Kenneth (A0081112Y)

Low Zhi Wei Kenny (A0086884J)

Group 8

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Table of Contents

1 Introduction 3

2 Objectives

2.1 Part I: Crystal Symmetry and Indexing 5

2.2 Part II: Precise Lattice Parameter Measurement by X-ray Diffraction 5

3 Experimental Procedure

2.1 Part I: Crystal Symmetry and Indexing 5

2.2 Part II: Precise Lattice Parameter Measurement by X-ray Diffraction 5

4 Results & Discussion for Part I

4.1 Tabulation of Data 8

4.2 Analysis of Results 9

5 Results & Discussion for Part II

5.1 Tabulation of Data 11

5.2 Analysis of Results 12

6. Sources of Errors 13

7. Conclusion 14

8. References 15

9. Appendix 16

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1. Introduction

X-Rays

X-rays are a form of electromagnetic radiation, with wavelengths ranging from 0.01 to 1 nanometers,

approximately the same size as an atom. In the electromagnetic spectrum, X-rays have shorter

wavelengths than ultraviolet (UV) rays, and longer wavelengths than gamma rays.

X-ray crystallography is used in two main areas, namely fingerprint characterisation of crystalline

materials, and determination of their structure.When a beam of X-ray strikes the crystal, it gets scattered

elastically into many specific directions, with its wavelength remaining unchanged. A three-dimensional

picture of the electron densities within the structure can be observed from the different angles and

intensities of these diffracted beams. Each crystalline solid has a unique set of patterns, which enables

identification of the material, and deduction of other information such as the mean positions of the atoms

in the crystal structure, their bonding, and disorder.

Generation of X-Rays

In this experiment, the X-rays were generated using a copper cathode X-ray tube. Electrons are liberated

from heating up the anode, and accelerated towards the cathode by high voltage. The electrons collide

with the atoms of the metal anode (or cathode? )causing excitation of electrons in the metal atoms. X-rays

are then produced when the electrons return to ground state (K level). As the different types of radiation

have different wavelengths, this results in difficulty in analyzing the diffraction patterns. Thus, only one

particular wavelength at Kα1 is used for analysis in diffraction experiments while while Kα2 is omitted and

Kβ radiation is usually filtered off,

In an X-ray diffraction measurement, a crystal is mounted on a goniometer and gradually rotated while

being bombarded with X-rays, producing a diffraction pattern of regularly spaced spots known as

reflections. The two-dimensional images taken at different rotations are converted into a three-

dimensional model of the density of electrons within the crystal using the mathematical method of Fourier

transforms, combined with chemical data known for the sample.

http://en.wikipedia.org/wiki/X-ray_crystallography

Crystal Symmetry and Indexing

A crystal structure consists of a three-dimensional periodic array of unit cells, made up of the same type

of atoms in equal spacings. Thus, a crystal structure is essentially a mathematical lattice defining the

position of the unit cell and the relative arrangement of atoms within the unit cell with respect to the

origin of the cell. As a result, it is sufficient to consider only the mathematical lattice when considering

the lattice constants, which leads to the assumption that there is one atom per unit cell. Any three-

dimensionally periodic lattice is considered to comprise of an infinite stack of parallel and equally spaced

planes.

Diffraction and Interference

Diffraction refers to the various phenomena that occurs when a wave meets an obstacle, while the term

interference refers to any situation in which two or more waves overlap in space. When a wavefront of X-

rays impinges on a periodic array of atoms, waves will be scattered at each atom.

http://en.wikipedia.org/wiki/Crystal

http://en.wikipedia.org/wiki/Goniometer

http://en.wikipedia.org/wiki/Fourier_transform

http://en.wikipedia.org/wiki/Fourier_transform

http://en.wikipedia.org/wiki/X-ray_crystallography

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http://webs.mn.catholic.edu.au/physics/emery/assets/9_4_fr9.gif

If the incident waves are in phase, the reflected waves will also be in phase and thus interfere

constructively after scattering at different atoms.

http://cdn4.explainthatstuff.com/waveinterference.gif

Constructive interference occurs when the path difference corresponds to an integral number of

wavelengths:

nλ = 2AB

nλ = 2 dhklsinθhkl

where λ is the wavelength, n is an integral number, dhkl is the interplanar spacing of the lattice, and θhkl

is the complement of the angle of incidence. This condition for constructive superposition of the

diffracted waves is known as Bragg’s Law.

http://webs.mn.catholic.edu.au/physics/emery/assets/9_4_fr9.gif

http://cdn4.explainthatstuff.com/waveinterference.gif

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In a polycrystalline sample, there are higher number of individual crystallites which are favourably

orientated to give rise to a diffracted beam in the direction of a detector, as compared to a single crystal,

provided that Bragg’s Law is satisfied for any of the dhkl,λ, and θhkl values.

X-Ray Powder Diffraction

Powder X-ray diffractogram reveals detailed information on crystal symmetry and lattice parameters, and

therefore, the crystalline phase, or the components of multiphase samples, can be identified. However,

determining the atomic position in the unit cell from the information available from single crystal

diffractometry is much easier than determining that from the information from powder diffractometry.

The instrument used is a powder X-ray diffractometer, and a set of diffraction peaks as a function of 2θ

will be recorded.

Precise Lattice Parameter Measurement by X-Ray Diffraction

For first order X-ray diffraction, n=1;

λ = 2 dhklsinθhkl

For a cubic system;

dhkl=a/√ (h2+k

2+l

2)

where a is the lattice parameter. Theoretically, the dhkl spacing, and thereafter, the lattice parameter a for a

cubic structure can be calculated when λ and θhkl is known. Precision in calculation of dhkl and a depends

on the precision of sinθhkl, and not θhkl. The error in sinθhkl caused by a given error in θhkl decreases as θhkl

increases, and thus, a very accurate value of sinθhkl can be obtained from a measurement of θhkl near 90º.

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2. Objectives

2.1 Part I: Crystal Symmetry and Indexing

This experiment aims to find out the Miller indices (hkl) for each peak of the given samples, iron powder

(Fe), Aluminum (Al) and Sodium Chloride (NaCl), and check if they lead to observable reflections hence

determining the crystal symmetry of each sample.

2.2 Part II: Precise Lattice Parameter Measurement by X-ray Diffraction

This experiment aims to determine the precise measurement lattice parameter a of Aluminum (Al) by

using X-Ray diffraction of sample at high angles, followed by calculation using Bragg’s Law,

(Equation 1)

and equation of a cubic system, as shown below.

(Equation 2)

3. Experimental Procedure

3.1 Part I: Crystal Symmetry and Indexing

1 Place each sample into an individual holder. Make sure that the sample is compact when placed to

ensure accuracy of data collected.

2 Mount the holders with samples carefully onto the diffractometer.

3 Using the software, set the parameters necessary for each sample.

4 Run the scanning.

5 For Aluminum sample, perform individual scanning for the angular range around each of the three

peaks. Make sure the measurement time/step is sufficiently high to reduce statistical errors in

determining peak position. Data values recorded will be used for Part II of the experiment.

6 Once the scanning is complete, obtain the recorded peak values using the software. Using the

software, filter out unnecessary peak values.

7 Calculate the Miller Indices (hkl) for each peak and record them in a table.

3.2 Part II: Precise Lattice Parameter Measurement by X-ray Diffraction

1 Use the data values obtained in Part I for the individual scanning around the 3 peaks.

2 Calculate the lattice parameter a for each peaks using Equations (1) and (2).

3 Plot a graph of a against cos2θ and extrapolate to where θ = 90°. Determine lattice parameter a for

Aluminum (Al).

4 Compare with theoretical values and determine the uncertainties of calculated values.

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4. Results & Discussion for Part I

4.1 Tabulation of Data

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Sodium Chloride (NaCl)

where a=dhkl*sqrt(h2+k

2+l

2)

Iron powder (Fe)

where a=dhkl*sqrt(h2+k

2+l

2)

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Aluminium (Al)

where a=dhkl*sqrt(h

2+k

2+l

2)

4.2 Analysis of Results

Based on the results obtained, the lattice parameter, a, is calculated with respect to the simple cubic

structure (SC), body-centered cubic structure (BCC) and face-centered cubic structure (FCC). After doing

so, the lattice parameter that trend linearly, that has the most similar values of lattice parameter will have

the characteristics of the particular cubic structure. This is required for lattice parameter should be

uniform throughout the particular specimen’s structure.

Sodium chloride: The table below shows that NaCl has a face-centered cubic structure.

Iron: The table

below shows

that Fe has a

body-centered cubic structure. This is based on contextual knowledge. However as shown in the table

below, iron has also a simple cubic structure. This could be due to the excitation of iron atoms using the

X-ray photons and the excited iron atoms remain at an energy level, known as the metastable state. This

state is only temporary and unstable. Therefore, the iron atoms start to exhibit a simple cubic structure.

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Aluminium: The table below shows that Al has a body-centered cubic structure.

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5. Results & Discussion for Part II

5.1 Tabulation of Data

(shifted below)

5.2 Analysis of Results

1. There are 2 conditions for further Bragg peaks:

a) sin θ ≤ 1, and

b) h,k,l are either all odd or even (due to Al’s FCC structure)

Based on this:

Let sin θ = 1 and Using λ(Kα1) and lattice parameter(a) of pure Al.

λ = 2dhklsin θ = 2dhkl

dhkl=a/√(h2+k

2+l

2)

(h2+k

2+l

2) = (a/dhkl)

2 = (2a/λ)

2 = 27.63 ≈ 27 (largest integer value)

As 42+2

2+2

2 = 24, we will consider planes where h

2+k

2+l

2 = 25,26,27

h2+k

2+l

2 Available Planes

25 (430),(500)

26 (431),(510)

27 (333),(511)

The next Bragg peak will be caused by the planes (333) and (511), as both fulfil the 2 conditions

mentioned above.

Finding 2θ for these 2 planes:

λ = 2[a/√(h2+k

2+l

2)] sin θ

θ = 81.27o

2θ = 162.54o

Theoretically, this Bragg peak will be detected by the Powder X-ray Diffractometer. However, as the

diffractometer can only scan up to a maximum of 160o, it will not be detected in the experiment. Hence,

(331), (420) and (422) are the 3 furthest Bragg peaks that can be used to determine the lattice constant.

2. (Inclusion of graphs)

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3.

where a=(λ*sqrt(h2+k

2+l

2)*10)/(2sinθhkl)

4.

5.

Experimental a = 4.0495Å

Theoretical a = 4.0494Å

% Error = 0.00246%

In this experiment, the obtained result was found to be very close to the actual lattice parameter of Al.

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6. Sources of Errors

It is important to note that despite the steps ensured to minimise errors in this experiment, inevitably there

is still systematic errors that may cause a slight deviation from the experimental values and the theoretical

values.

Although a powdered and flat samples are used in this experiment to achieve a right orientation, it is often

hard to ensure that the powdered samples are in the right orientation perfectly for diffraction to occur.

Hence the non-flat surface of the sample may diffract the X-ray with a slight deviation in the θ, due to

the change of angle of incidence rays. As the theoretical standard lattice parameter of the sample

was probably obtained from single crystal X-Ray diffraction, the using of powder X-Ray

diffraction may be less precise than that of a single crystal, hence resulting in a inaccuracy of

results.

In the non-zero Kelvin temperature, the moving atoms of the sample will indirectly cause a shift

in the 2θ values obtained. Hence it will not be at a ideal position to form a plane that accounts

accurately for the peaks obtained in the graph. Furthermore, as instead of elastic collisions,

inelastic collisions may happen when the X-ray hits the crystal structure of the sample. Hence the

angle of reflected rays may not be equal to incident ray, resulting in the inaccuracy of the angle

2θ.

Also, the aluminium sample may contain impurities that may affect the crystal structure of the material

which in turns may affect the lattice parameter deduced. As aluminium is a reactive metal, it may be

oxidized into Al2O3 when it is exposed in air. Hence causing systematic errors.

As X-rays can be absorbed by protein and water in the crystal, uneven absorption of the X-rays by the

sample can lower the intensities which will alter the positions of peaks and affect accuracy of results.

Other than the errors derived from the sample, systematic errors can also be due to the setup of the

experimental apparatus. For example, the close proximity between the X-Ray source and the detector may

cause the detection of the stray X-Rays reflected, hence causing the inaccuracy of the peaks obtained. In

the event that the X-ray are able to reach the sample holder, any non-zero background sample holder used

may also produce X-Ray diffraction peaks as such holder may exhibit some crystal structure as well.

Hence causing systematic errors that affects the peaks in the graphs obtained.

7. Conclusion

X-ray diffraction allows us to find out the cell structure of various crystalline compounds. We concluded

that NaCl and Al have FCC structures, while Fe has a BCC structure. This can be deduced from the

Bragg’s Law by comparing the relation between the angles where Bragg peaks are detected. However

there are certain limitation to the XRD being use as an identification technique where contextual

knowledge is still required in the determination of Fe structure. Such limitations include the inability of

the X-ray generator and the detector to form an 2θ angle greater than 160o.

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8. References

Uncertainty Estimation of Lattice Parameters measured by X-Ray Diffractions. (2006). Retrieved from

http://www.imeko.org/publications/wc-2006/PWC-2006-TC8-008u.pdf

Elements of X-ray diffraction, by B.D. Cullity and S.R. Stock, 3rd Edition, Pearson

Education International / Prentice Hall, 2001.

G.S. Rohrer, Structure and bonding in Crystalline Materials, Cambridge University

Press, 2001.

http://www.imeko.org/publications/wc-2006/PWC-2006-TC8-008u.pdf

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9. Appendix

Figure 1. Plot of Intensity vs 2θ (111.500° - 112.976°) (Aluminum)

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Figure 4. Full plot of Intensity vs 2θ (Aluminum)

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Figure 5. Full plot of Intensity vs 2θ (Iron Powder)

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Figure 6. Full plot of Intensity vs 2θ (Sodium Chloride)

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