Experimentally, the Bragg law can be applied in two different ways: Experimentally, the Bragg law can be applied in two different ways:

By using x-rays of known By using x-rays of known and measuring the angle and measuring the angle we can we can determine the spacing determine the spacing dd of various planes in a crystal: this is called of various planes in a crystal: this is called structure analysisstructure analysis (X-ray diffraction or XRD) (X-ray diffraction or XRD)

Alternatively, we can use a crystal with planes of known spacing Alternatively, we can use a crystal with planes of known spacing dd, , measure measure and thus determine the wavelength and thus determine the wavelength of the radiation: of the radiation: this is called this is called x-ray spectroscopy (x-ray fluorescencex-ray spectroscopy (x-ray fluorescence)) (XRF) (XRF)

• The instrument for studying materials by measurements of the way in The instrument for studying materials by measurements of the way in which they diffract x-rays of known wavelength which they diffract x-rays of known wavelength is called the is called the diffractometerdiffractometer. .

• The x-ray diffractometer is the most important tool for performing The x-ray diffractometer is the most important tool for performing diffraction analysis of materials.diffraction analysis of materials.

• In an x-ray diffractometer, the film of a powder camera In an x-ray diffractometer, the film of a powder camera (Debye-(Debye-Scherrer)Scherrer) is replaced by a movable counter. is replaced by a movable counter.

• All diffractometers have the following components:All diffractometers have the following components:

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X-ray sourceX-ray source

X-ray beam conditioning devicesX-ray beam conditioning devices

Sample and detector rotationSample and detector rotation

Radiation detector and associated electronicsRadiation detector and associated electronics

Control and data acquisition system.Control and data acquisition system.

• The components are arranged about a circle (diffractometer circle) The components are arranged about a circle (diffractometer circle) which lies in a plane called the diffractometer plane.which lies in a plane called the diffractometer plane.

• Both the x-ray source and the detector lie on the circumference of the Both the x-ray source and the detector lie on the circumference of the circle.circle.

• The angle between the plane of specimen and x-ray source is = The angle between the plane of specimen and x-ray source is = , the , the Bragg angle.Bragg angle.

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Figure 1: X-ray Figure 1: X-ray Diffractometer (schematic)Diffractometer (schematic)

T

C

F

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Figure 1: X-ray Diffractometer (Siemens at UTM)Figure 1: X-ray Diffractometer (Siemens at UTM)

X-ray Tube

Counter

SampleReceiving

Slits

Diverging Slits

2

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Figure 2Figure 2

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• The angle between the projection of x-ray source and the detector is = 2 .

• The x-ray source is fixed, and the detector moves through a range of angles.

• The radius of the focusing circle is not constant but increases as the angle 2 decreases. The 2 measurement is typically from 0o to about 170 o.

• The choice of range depends on the crystal structure of the material (if known) and the time you want to spend obtaining the diffraction pattern

• For an unknown specimen a large range of angles is often used because the positions of the reflections are known, at least not yet!

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• The incident x-ray beam is defined by a set of Soller slits (The incident x-ray beam is defined by a set of Soller slits (divergence divergence slitsslits), Figure 2), Figure 2

• The sample sits at the centre of the diffractometer.The sample sits at the centre of the diffractometer.

• The Bragg beam from the sample is defined by a second set of slits The Bragg beam from the sample is defined by a second set of slits ((receiving slitsreceiving slits))

• The slits (divergence and receiving) consist of a series of closely The slits (divergence and receiving) consist of a series of closely spaced parallel metal plates that define and collimate the incident x-ray spaced parallel metal plates that define and collimate the incident x-ray beam (make the beam parallel)beam (make the beam parallel)

• The slits are typically about 30 mm long and 0.05 mm thick, and the The slits are typically about 30 mm long and 0.05 mm thick, and the distance between them is about 0.5 mm.distance between them is about 0.5 mm.

• The slits are usually made of a metal with a high atomic number such The slits are usually made of a metal with a high atomic number such as Mo, or Ta (because of their high absorption capacities) as Mo, or Ta (because of their high absorption capacities)

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OPERATION OF A DIFFRACTOMETEROPERATION OF A DIFFRACTOMETER

• From Figure 1, x-rays are produced at the target, From Figure 1, x-rays are produced at the target, TT, of the x-ray tube., of the x-ray tube.

• These x-rays are usually filtered to produce monochromatic radiation, These x-rays are usually filtered to produce monochromatic radiation, collimated (to produce a beam composed of perfectly parallel rays) and collimated (to produce a beam composed of perfectly parallel rays) and then hit the specimen at then hit the specimen at CC..

• The x-rays diffracted by the specimen are focused through a slit The x-rays diffracted by the specimen are focused through a slit FF onto onto the counter. the counter.

• As the counter moves on the diffractometer circle through an angle 2As the counter moves on the diffractometer circle through an angle 2, , the specimen rotates through an angle the specimen rotates through an angle . .

• The x-ray quanta are converted into electrical pulses by an x-ray The x-ray quanta are converted into electrical pulses by an x-ray detector. The detector output is sent to a counterdetector. The detector output is sent to a counter

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• The counter counts the number of current pulses / unit time, and this The counter counts the number of current pulses / unit time, and this number is directly proportional to the intensity (energy) of the diffracted number is directly proportional to the intensity (energy) of the diffracted x-ray beam.x-ray beam.

• A typical diffraction pattern A typical diffraction pattern for aluminium for aluminium produced in a diffractometer produced in a diffractometer is shown in Figure 3is shown in Figure 3

• There is fundamental difference between the operation of a powder There is fundamental difference between the operation of a powder camera and a diffractometer. camera and a diffractometer.

• In a powder camera, all diffraction lines are recorded simultaneously In a powder camera, all diffraction lines are recorded simultaneously and variations in the intensities of the incident x-ray beam during and variations in the intensities of the incident x-ray beam during analysis can have no effect of the relative line intensities in the analysis can have no effect of the relative line intensities in the diffraction pattern.diffraction pattern.

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Figure 3: X-ray diffraction pattern of Aluminium

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• With a diffractometer, diffraction lines are recorded one after the other, With a diffractometer, diffraction lines are recorded one after the other, and thus, it is imperative to keep the incident beam constant when and thus, it is imperative to keep the incident beam constant when relative intensities are measured.relative intensities are measured.

EXAMINATION OF A STANDARD X-RAY DIFFRACTION PATTERNEXAMINATION OF A STANDARD X-RAY DIFFRACTION PATTERN

• An example of a typical x-ray diffraction pattern of Aluminium is shown An example of a typical x-ray diffraction pattern of Aluminium is shown in Figure in Figure 33

• The pattern consists of a series of peaks, which are also called The pattern consists of a series of peaks, which are also called reflections. The peak intensity is plotted vs. measured diffraction angle, reflections. The peak intensity is plotted vs. measured diffraction angle, 22. .

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• Each peak corresponds to x-rays diffracted from a specific set of Each peak corresponds to x-rays diffracted from a specific set of planes in the specimen, and these peaks are of different heights planes in the specimen, and these peaks are of different heights (intensities).(intensities).

• The intensity is proportional to the number of x-rays of a particular The intensity is proportional to the number of x-rays of a particular energy that have been counted by the detector for each angle 2energy that have been counted by the detector for each angle 2. .

• For diffraction analysis, we use the relative intensities of peaks For diffraction analysis, we use the relative intensities of peaks because measuring absolute intensity is very difficult.because measuring absolute intensity is very difficult.

• The position of the peaks in an x-ray diffraction pattern depends on the The position of the peaks in an x-ray diffraction pattern depends on the crystal structure (shape and size of the unit cell) of the material.crystal structure (shape and size of the unit cell) of the material.

• For low values of 2For low values of 2 each reflection appears as a single peak. For each reflection appears as a single peak. For higher values of 2higher values of 2 each reflection consists of a pair (two) peaks, each reflection consists of a pair (two) peaks, which correspond to diffraction of the Kwhich correspond to diffraction of the K11 and K and K22 wavelengths. wavelengths.

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Figure 5: Comparison of x-ray diffraction patterns from cubic materials

Diffraction patterns from cubic materials can usually be Diffraction patterns from cubic materials can usually be distinguished from those of non-cubic materials. distinguished from those of non-cubic materials.

Figure 5 shows the calculated diffraction patterns of the four cubic Figure 5 shows the calculated diffraction patterns of the four cubic crystal structures.crystal structures.

110 200 211 220 310 222 321

111 200

111

220

220

311

311

222

400

400

BCCBCC

FCCFCC

DiamondDiamond

SCSC

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Forbidden Reflections

h2+k2+l2 Primitive cubic Face-centred cubic Body-centred cubic1 100 - -2 110 - 1103 111 111 -4 200 200 2005 210 - -6 211 - 2117 - - -8 220 220 2209 221/300 - -10 310 - 31011 311 311 -12 222 222 22213 320 - -14 321 - 32115 - - -16 400 400 400

Primitive Cubic FCC BCCh2 + k2 + l2

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INDEXING THE DIFFRACTION PATTERNINDEXING THE DIFFRACTION PATTERN

• Knowing the crystal structure of a material is central to understanding Knowing the crystal structure of a material is central to understanding the behaviour of materials under stress, alloy formation and phase the behaviour of materials under stress, alloy formation and phase transformations.transformations.

• The size and shape of the unit cell determine the angular positions of The size and shape of the unit cell determine the angular positions of the diffraction peaks, and the arrangement of the atoms within the unit the diffraction peaks, and the arrangement of the atoms within the unit cell determines the relative intensities of the peaks.cell determines the relative intensities of the peaks.

• It is therefore possible to calculate the size and shape of the cell from It is therefore possible to calculate the size and shape of the cell from the angular positions of the peaks and the atom positions in the unit the angular positions of the peaks and the atom positions in the unit cell from the intensities of the diffraction peaks.cell from the intensities of the diffraction peaks.

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• Indexing the pattern involves assigning the correct Miller indices to Indexing the pattern involves assigning the correct Miller indices to each peak in the diffraction patterneach peak in the diffraction pattern

• It is important to remember that correct indexing is done only when all It is important to remember that correct indexing is done only when all the peaks in the diffraction pattern are accounted for and no peaks the peaks in the diffraction pattern are accounted for and no peaks expected for the structure are missing from the patternexpected for the structure are missing from the pattern

• An example of indexing a pattern from a material with a cubic structure An example of indexing a pattern from a material with a cubic structure is presented here. is presented here.

• Two methods can be used to index a diffraction a patternTwo methods can be used to index a diffraction a pattern

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METHOD 1: Diffraction will occur when Bragg law is satisfied:Diffraction will occur when Bragg law is satisfied:

The interplanar spacing d for a cubic material is given by:The interplanar spacing d for a cubic material is given by:

Combining the above equations results in:Combining the above equations results in:

sin2d

222 lkh

adhkl

2

2

2222

sin4

lkh

ad

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Which gives:Which gives:

Since Since 22 / 4a / 4a22 is constant, sin is constant, sin 22 is proportional to (h is proportional to (h22 + k + k22 + l + l22),),

As As increases, planes with higher Miller indices will diffract. increases, planes with higher Miller indices will diffract.

Writing the above equation for two different planes and diving by the Writing the above equation for two different planes and diving by the minimum plane, we get:minimum plane, we get:

2222

22

4lkh

aSin

22

22

22

21

21

21

221

2

sin

sin

lkh

lkh

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Example: indexing of Aluminium diffraction pattern by method 1Example: indexing of Aluminium diffraction pattern by method 1

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1. Identify the peaks

2. Determine sin2

3. Calculate the ratio sin2 / sin2min and multiply by the

appropriate integers (1, 2, or 3)

4. Select the result from step (3) that yields h2 + k2 + l2 as an

integer

5. Compare results with the sequences of h2 + k2 + l2 values to

identify the Bravais lattice

6. Calculate lattice parameter.

Example: indexing of Aluminium diffraction pattern by method 1Example: indexing of Aluminium diffraction pattern by method 1

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• The bravais lattice can be identified by noting the systematic presence The bravais lattice can be identified by noting the systematic presence (or absence) of reflections in the diffraction pattern.(or absence) of reflections in the diffraction pattern.

• The Table below illustrates the selection rules for cubic lattices. The Table below illustrates the selection rules for cubic lattices.

• According to these rules, the (hAccording to these rules, the (h22 + k + k22 + l + l22) values for the different cubic ) values for the different cubic lattices follow the sequence:lattices follow the sequence:

Simple cubic : 1,2,3,4,5,6,8,9,10,11,12,13,14,16,….

BCC : 2,4,6,8,10,12,14,16,18,...

FCC : 3,4,8,11,12,16,19,20,24,27,32,…

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Bravais latticeBravais lattice Reflections present forReflections present for Reflections absent forReflections absent for

Primitive (simple cubic)Primitive (simple cubic) All None

Body Centered Cubic Body Centered Cubic (BCC)(BCC)

h + k + l = even h + k + l = odd

Face Centered Cubic Face Centered Cubic (FCC)(FCC)

h, k, l = unmixed (all even or all odd)

h, k, l = mixed

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CALCULATION OF THE LATTICE PARAMETERCALCULATION OF THE LATTICE PARAMETER

The lattice parameter,a, can be calculated from:The lattice parameter,a, can be calculated from:

Rearranging givesRearranging gives

2222

22

4lkh

aSin

2222

22

sin4lkha

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METHOD 2:METHOD 2:

This method can be used to index This method can be used to index the the diffraction pattern from materials with a diffraction pattern from materials with a cubic structure. From:cubic structure. From:

Since Since 22 / 4a / 4a22 is constant for any pattern and which we will call is constant for any pattern and which we will call AA, we ca, we cann write:write:

2222

22

4lkh

aSin

2222sin lkhA

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In a cubic system, the possible (hIn a cubic system, the possible (h22 + k + k22 + l + l22) values are: 1,) values are: 1, 2,2, 3,3, 4,4, 5,5, 6,6, 8, …. (even though all may not be present in every type of cubic 8, …. (even though all may not be present in every type of cubic lattice).lattice).

The observed sinThe observed sin22 values for all peaks in the pattern are therefore values for all peaks in the pattern are therefore divided by the integers divided by the integers 1, 1, 2, 3, 4, 5, 6, 8, to obtain a common quotient, 2, 3, 4, 5, 6, 8, to obtain a common quotient, which is the value of A, corresponding to (hwhich is the value of A, corresponding to (h22 + k + k22 + l + l22) =1.) =1.

We can then calculate the lattice parameter from the value of A using We can then calculate the lattice parameter from the value of A using the relationship:the relationship:

Aaor

aA

24 2

2

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Note that 0.1448 is also common in 1, 2, 3, 4, 5, 6, BUT absent in 8

It can only be FCC1-30