X-Ray Methods for Analysis of Materials

8/6/2019 X-Ray Methods for Analysis of Materials

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Text Books- Instrumental Methods of Analysis- Atkins Physical Chemistry

Lecture notes, not really enough but with somebackground reading should be fine for theexam!

Introduction X-ray diffraction (XRD)

Crystal structureX-ray fluorescence analysis (XRF, EDX)X-ray photoelectron spectroscopy (XPS)

Topics


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Traditional methods of analysis such as:Atomic Adsorption,Mass Spec,Chromatography,Infra-red spectroscopy,UV-Vis spectroscopy etc require dissolution into a fluid phase: destructive

Materials or solids analysis has been driven by the requirement to produce non-destructive methods.

They can be described in six categories:-Elemental the atoms present XRF, EDXStructural define atomic arrangement - XRDChemical define the chemical state - XPSImaging what does the morphology look like? Electron MicroscopySpectroscopic energy level transitions IRThermal effects of heating (sorption) TGA, DSC

Problems:- DestructiveElementalNo understanding of theirform in the material

INTRODUCTION


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W

hy to study solid state? Solid state includes most of the materials inthat make modern technology possible

Properties of the solid state differ significantlyfrom the properties of isolated atoms ormolecules

The term structure takes on a whole newmeaning.

Example: Nanosized metalparticles

Consider complete delocalizedelectrons in the metals, specialeffect called plasmons

(quantized plasma oscillations,collective oscillations of theelectron cloud) result is specificcolour of metal particles withnanosized dimensions.


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Challenges in analysis of solidscompared to molecular analysis

1. Solids have continuum energy states compared to discrete energylevels in molecular sates.

2. Very high absorptions so that not much energy gets out! Can lead toadsorption of signal of some elements within the matrix!

3. Saturation effects can not be simply diluted out.

4. Structural differences can often be difficult to resolve.

5. Many of the probes can not be used in simple laboratoryenvironments.

6. Require complex detection equipment.

7. Standards can be quire difficult to prepare compared to simpledilution. Phase diagrams and structural change. Matrix effects.


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Interaction of the radiation with the matter.

Elastic interactions - in which there is no lost in the energy

Inelastic interactions with lost of the exciting energy.


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Examples of Methods and Interactions with solids

X-ray Diffraction X-rays in and out but no loss of energy, elastic scattering of x-raysX-ray fluorescence X-rays in and out but look at x-rays generated by secondaryprocess, inelastic scattering of x-raysX-ray Absorption Spectroscopy - X-rays in and out but look for energy lossesX-ray Photoelectron Spectroscopy - X-rays in electrons out (secondary electrons)

Electron microscopy electrons in and out analyse either transmitted (transmissionEM) or secondary (secondary electrons, inelastic scattering of electrons)Low Energy Electron Energy Loss Spectroscopy - electrons in and out energyanalyse the electrons to look at energy loss to give vibrational information

Ultrasound - Sound in and out use it to analyse for void formation in solidsNMR radio frequency in and out analyse for energylossNo single techniques is capable of providing a complete characterization of a

solid


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Bulk vs Surface Analysis

Analysing solids is further complicated by

the depth sensitivity.

X-rays deeply penetrate matter so theanalyte depth is high and the analysis isbulk sensitive rather than surfacesensitive.

However, techniques involving electrons(even if they are excited by high energytechniques) are strongly absorbed and sooriginate from the surface region. Such areEDX and XPS.

This can be an advantage since surfacechemistry is of fundamental importance.However, phase separation andsegregation can give rise to problems.

Quantification

An ideal technique should be

quantitative e.g.

Signal height is proportional to thenumber of atoms of element orchemical state present.

This is rare in a solid. In molecularanalysis the intensity is usually directlyproportional to the number of moleculesin the analyte. This is because thesystems are frequently dilute there is

little chance that there are multipleinteractions of other interactions withprimary or secondary radiation. In asolid exactly the opposite is true. Thus,solids suffer matrix effects.


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Analyte

Matrix

Weak emitter in a strong adsorbingmatrix. Analyte emission absorbed bymatrix and no signal leaves sample.

Weak emitter as a thin layer at the

surface on strongly absorbing matrix.


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Most of the electrons in the incident beam lose energy upon entering a materialthrough inelastic scattering or collisions with other electrons of the material andform heat.

In such a collision the momentum transferfrom the incident electron to an atomicelectron can be expressed as dp = 2e2 / bv, where b is the distance of closestapproach between the electrons, and vis the incident electron velocity.

The energy transferred by the collision is given by T= (dp)2 / 2m = e4 / Eb2, wherem is the electron mass and Eis the incident electron energy, given by E= (1 /2)mv2.

By integrating over all values ofTbetween the lowest binding energy, Eo, and theincident energy E, one obtains the result that the total cross section for collision isinversely proportional to the incident energy E.

How is heat produced ?


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A second type of interaction in whichthe incident electron can lose itskinetic energy is an interaction withthe nucleus of a target atom. In thistype of interaction, the kinetic energyof the projectile electron is convertedinto electromagnetic energy.


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Interaction of X-rays with matter


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A single crystal, also calledmonocrystal, is a crystalline solid inwhich the crystal lattice of the entiresample is continuous and unbroken tothe edges of the sample, with no grainboundaries.

The opposite of a single crystalsample is apolycrystallinesample, which is made up of anumber of smaller crystals

known as crystallites. Usuallythose crystallites are connectedthrough a amorphous material toform extended solid.


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incorrect


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Symmetry Operations and Elements

A Symmetry operation is an operation that can be performed eitherphysically or imaginatively that results in no change in the

appearance of an object.

There are 3 types of symmetry operations: rotation, reflection, andinversion. .

Rotational SymmetryAs illustrated above, if an object can be rotated about an axis andrepeats itself every 90o of rotation then it is said to have an axis of 4-fold rotational symmetry. The axis along which the rotation isperformed is an element of symmetry referred to as a rotationaxis. The following types of rotational symmetry axes are possible in

crystals.


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1-Fold Rotation Axis - Anobject that requires rotation ofa full 360o in order to restore itto its original appearance has

no rotational symmetry. Sinceit repeats itself 1 time every360o it is said to have a 1-foldaxis of rotational symmetry.

2-fold Rotation Axis- If an

object appears identical after arotation of 180o, that is twice ina 360o rotation, then it is said tohave a 2-fold rotation axis(360/180 = 2). Note that inthese examples the axes we are

referring to are imaginary linesthat extend toward youperpendicular to the page orblackboard. A filled oval shaperepresents the point where the2-fold rotation axis intersects

the page.


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3-Fold Rotation Axis- Objects that repeatthemselves upon rotation of 120o are saidto have a 3-fold axis of rotational symmetry

(360/120 =3), and they will repeat 3 timesin a 360o rotation. A filled triangle is usedto symbolize the location of 3-fold rotationaxis.

4-Fold Rotation Axis - If an object

repeats itself after 90o of rotation, it willrepeat 4 times in a 360o rotation, asillustrated previously. A filled square isused to symbolize the location of 4-foldaxis of rotational symmetry.

6-Fold Rotation Axis - If rotation of 60oabout an axis causes the object to repeatitself, then it has 6-fold axis of rotationalsymmetry (360/60=6). A filled hexagon isused as the symbol for a 6-fold rotationaxis.


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Mirror Symmetry

A mirror symmetry operation is an imaginary operation that can be performed toreproduce an object. The operation is done by imagining that you cut the object in half,

then place a mirror next to one of the halves of the object along the cut. If the reflectionin the mirror reproduces the other half of the object, then the object is said to have mirrorsymmetry. The plane of the mirror is an element of symmetry referred to as a mirror

plane, and is symbolized with the letterm. As an example, the human body is an objectthat approximates mirror symmetry, with the mirror plane cutting through the center ofthe head, the center of nose and down to the groin.


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Center of Symmetry

Another operation that can beperformed is inversion through apoint. In this operation lines aredrawn from all points on the objectthrough a point in the center of theobject, called a symmetry center(symbolized with the letter "i"). Thelines each have lengths that areequidistant from the original

points. When the ends of the linesare connected, the original object isreproduced inverted from its originalappearance. In the diagram shownhere, only a few such lines are drawnfor the small triangular face. The right

hand diagram shows the objectwithout the imaginary lines thatreproduced the object.


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Combinations of Symmetry Operations


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y y p

As should be evident by now, in three dimensional objects,such as crystals, symmetry elements may be present in severaldifferent combinations. In fact, in crystals there are 32 possiblecombinations of symmetry elements. These 32 combinations

define the 32Point Groups.

Thus, this crystal has thefollowing symmetry elements:1 - 4-fold rotation axis (A4)

4 - 2-fold rotation axes (A2), 2cutting the faces & 2 cutting theedges.5 mirror planes (m), 2 cuttingacross the faces, 2 cuttingthrough the edges, and one

cutting horizontally through thecenter.Note also that there is a centerof symmetry (i).The symmetry content of thiscrystal is thus: i, 1A4, 4A2, 5m


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The space groups in three dimensions are made from combinations of the 32


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Crystal family Crystal systemRequiredsymmetries ofpoint group

point groups space groups bravais lattices Lattice system

Triclinic None 2 2 1 Triclinic

Monoclinic1 twofold axisof rotation or 1mirror plane

3 13 2 Monoclinic

Orthorhombic

3 twofold axesof rotation or 1twofold axis ofrotation and twomirror planes.

3 59 4 Orthorhombic

Tetragonal 1 fourfold axisof rotation

7 68 2 Tetragonal

HexagonalTrigonal

1 threefold axisof rotation

57 1 Rhombohedral18

1 HexagonalHexagonal

1 sixfold axis ofrotation

7 27

Cubic4 threefold axes

of rotation

5 36 3 Cubic

Total: 6 7 32 230 14 7

crystallographic point groups with the 14 Bravais lattices which belong to one of 7lattice systems. This results in a space group being some combination of thetranslational symmetry of a unit cell including lattice centering, the point groupsymmetry operations ofreflection, rotation and improper rotation (also called

rotoinversion), and the screw axis and glide plane symmetry operations. Thecombination of all these symmetry operations results in a total of 230 unique spacegroups describing all possible crystal symmetries


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Double Slit and Diffraction Grating.


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a pattern ofdark andbrightfringes

At C all wavelengths arrivein phase and interfere

constructively to produce acentral image

At some other point P which is at a distanceL from one slit and L + n from the other ( is

some specific wavelength present in the lightbeam; n is an integer) there is alsoconstructive interferences and a bright fringeappears with the color pertaining to thatspecific wavelength. At intermediate pointsdistant L and L + (2n + 1) (/2), destructiveinterference occurs for that wavelength .


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Another view of Braggs law


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Another view of Bragg s law

When an X-ray beam hits an atom, the electrons around the atom start tooscillate with the same frequency as the incoming beam. In almost alldirections we will have destructive interference. However the atoms in acrystal are arranged in a regular pattern, and in a very few directions wewill have constructive interference. Hence, a diffracted beam may bedescribed as a beam composed of a large number of scattered raysmutually reinforcing one another.


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vector product


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X-ray diffraction has been in use in two main areas:

1) Fingerprint characterization of crystalline materials powder XRD

2) The determination of crystal structure e.g. identification ofatomic position of crystals - single crystal XRD analysis.

But also1) Strain and lattice mismatch in crystals2) Orientation of thin films grazing incidence XRD, rocking

curves3) Low angle X-ray scattering form and shape of polymers,

meso-materials, colloidal particles.


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Single Crystal Polycrystalline


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Each crystalline solid has its unique characteristic X-ray powder pattern

which may be used as a "fingerprint" for its identification. Once the materialhas been identified, X-ray crystallography may be used to determine itsstructure, i.e. how the atoms pack together in the crystalline state and whatthe interatomic distance and angle are etc. X-ray diffraction is one of themost important characterization tools used in solid state chemistry andmaterials science. Every lab in materials science has an XRD instrument.

Powder XRD


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In any real solid we have a chance orientation. It would be almost impossible tostudy diffraction. However, 99% of all materials are polycrystalline or can be

prepared (by grinding) so as to present many grains of material. In these some willalways be at the correct alignment.

Bragg-Brentano Geometry ( - 2)


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sample

source

detector

Bragg Brentano Geometry ( 2)


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Divergence slitAntiscatter slit

Monochromator

Detector slit

Tube

soller slit

Which Information does a Powder


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dimension of the elementary cellcontent of the elementary cellstrain/crystallite sizequantitative phase amount

Which Information does a PowderPattern offer?

1) peak position d-spacing2) peak intensity structure factors3) peak broadening size of a crystallite

Primary information

Secondary information - calculated

In the kinematical approximation for diffraction, the intensity of a diffracted beam is


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In the kinematical approximation for diffraction, the intensity of a diffracted beam isgiven by:

where is the wavefunction of a beam scattered a vector ,

and is the so called structure factor which is given by:

Here, rj is the position of an atom j in the unit cell, and fj is the scattering power ofthe atom, also called the atomic form factor. The sum is over all atoms in the unitcell. It can be shown that in the ideal case, diffraction only occurs if the scatteringvector is equal to a reciprocal lattice vector .

The structure factor describes the way in which an incident beam is scattered by theatoms of a crystal unit cell, taking into account the different scattering power of theelements through the term fj. Since the atoms are spatially distributed in the unit cell,there will be a difference in phase when considering the scattered amplitude from

two atoms. This phase shift is taken into account by the complex exponential term.The atomic form factor, or scattering power, of an element depends on the type ofradiation considered. Because electrons interact with matter though differentprocesses than for example X-rays, the atomic form factors for the two cases are notthe same.


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X li b d i


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When x-rays enter asolid they undergorefraction. For x-raysthis is very small.

But the refraction angle differs from the incidence angle by only parts perthousand. But because of this the path length difference slowly variesfrom planes deeper into the material. The constructive interferenceslowly becomes destructive.Provided the sample is thick enough (if a crystallite is 10 m there are 1 x

10-5/10-10 atom planes = 105) then all these slightly out of phasereflections will cancel. Leave just the perfect constructive interferencefeature. But if the crystallite is small , below 100 nm there are only 1000planes and the effect of canceling is not so pronounced which will lead tobroadening of the reflection.

X-ray line broadening


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S t ti b


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Some types of unit cell give characteristic and easily recognizable patternlines. For example, in a cubic primitive lattice of unit cell dimensions a thespacing is given by the equation:

sin = (h2 + k2 + l2)1/2 /2a

The reflections arte than predicted by substituting the values h, k, l:{h,k,l} {100} {110} {111} {200} {210} {211} {220}h2 + k2 + l2 1 2 3 4 5 6 8

The 7 (and 15) is missing because the sum of the squares of threeintegers cannot be 7. Therefore the pattern has systematic absences thatare characteristic of the cubic P lattice.

Systematic absences

For FCC, h,k,l all even or all odd are presentFor BCC, h+k+l = odd are absent

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X-Ray Methods for Analysis of Materials